Answer:
Step-by-step explanation:
Final answer:
The decision rule for the hypothesis test at a 0.10 significance level involves rejecting the null hypothesis if the test statistic is greater than the critical value. The calculated Z-value is approximately 1.8233, and the corresponding p-value will determine our decision to reject or not reject the null hypothesis.
Explanation:
Decision Rule and Test Statistic
For a hypothesis test comparing two population means with known standard deviations, we use a Z-test for two samples. The decision rule at a significance level of 0.10 involves rejecting the null hypothesis (H0: μ1 = μ2) if the absolute value of the test statistic exceeds the critical value from the standard normal distribution.
Calculation of Test Statistic
The test statistic (Z) is calculated using the formula: Z = (101.0 - 99.3)/ √[ (4.6²/40) + (4.0²/47) ]
Plugging in the numbers: Z = (1.7)/ √[ (21.16/40) + (16/47) ] Z = (1.7)/ √[ 0.529 + 0.3404255 ] Z = (1.7)/√[0.8694255] Z = (1.7)/0.932447 Z ≈ 1.8233
Decision Based on P-value
The p-value associated with this Z-value can be found using standard normal distribution tables or software. If the p-value is less than 0.10, we reject the null hypothesis. Otherwise, we fail to reject it.
Water is being pumped continuously from a pool at a rate proportional to the amount of water left in the pool. Initially there was 15,000 gallons of water in the pool; six minutes later there was 13,800 gallons.
At what rate was the amount of water in the pool decreasing when there were 14,000 gallons remaining and when will there be 5,000 gallons remaining?
Please show all steps.
Answer:
194.6 gpm at 14,000 gallons69.5 gpm at 5,000 gallons.Step-by-step explanation:
When a value is decreasing at a rate proportional to that value, it can be modeled by the formula
a = a0·e^(-kt)
where k is the constant of proportionality.
Alternatively, we can write the exponential function describing the pool volume* as ...
a = 15000·(138/150)^(t/6) = 15000·((138/150)^(1/6))^t
Comparing these, we see that ...
e^(-kt) = (138/150)^(t/6)
or ...
k = -ln(138/150)/6 ≈ 0.0138969
__
So, when 14000 gallons remain, the rate of decrease is ...
14000·0.0138969 ≈ 194.6 . . . gallons per minute
When 5000 gallons remain, the rate of decrease is ...
5000·0.0138969 ≈ 69.5 . . . gallons per minute
_____
* The generic form of this is ...
(initial value) · (multiplier per interval)^(number of intervals)
Here, the multiplier over a 6-minute period is 13800/15000 = 138/150, and the number of 6-minute intervals is t/6 when t is in minutes.
_____
Effectively, we make use of the fact that for ...
a = a0·e^(-kt)
the derivative is ...
da/dt = -k(a0·e^(-kt)) = -k·a
That is, k is the constant of proportionality mentioned in the first sentence of the problem statement.
Design specifications for filling a bottled soda claim that bottles should contain 350-360 milliliters of liquid. Sample data indicate that the bottles contain an average of 355 milliliters of liquid, with a standard deviation of 2 milliliters. Is the filling operation capable of meeting the design specifications?
Answer:
It is high likely that the filling operation is capale of meeting design specifications.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 355, \sigma = 2[/tex]
Is the filling operation capable of meeting the design specifications?
It will be capable if it is highly likely that the specifications will be met. A probability is said to be high likely when it is of at least 95%.
In this case, the probability of containing between 350 and 360 ml of liquid is the pvalue of Z when X = 360 subtracted by the pvalue of Z when X = 350.
X = 360
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{360 - 355}{2}[/tex]
[tex]Z = 2.5[/tex]
[tex]Z = 2.5[/tex] has a pvalue of 0.9938.
X = 350
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{350 - 355}{2}[/tex]
[tex]Z = -2.5[/tex]
[tex]Z = -2.5[/tex] has a pvalue of 0.0062.
This means that there is a 0.9938 - 0.0062 = 0.9876 = 98.76% probability that the filling operation is capable of meeting the design specifications. It is high likely that the filling operation is capale of meeting design specifications.
Find the area of the following regions, expressing your results in terms of the positive integer n ≥ 2. The region bounded by f(x)=x and g(x)=x^1/n, for x≥0
Answer:
The area of the searched region is [tex]A= a+b+ \frac{2n}{n+1}- \frac{n(a^{\frac{n+1}{n} }+b^{\frac{n+1}{n} }) }{n+1}-2[/tex]
Step-by-step explanation:
If you want to find the area of a region bounded by functions f(x) and G(x) between two limits (a,b), you have to do a double integral. you must first know which of the functions is greater than the other for the entire domain.
In this case, for 0<x<1, f(x)<g(x)
while for 1<x, g(x)<f(x).
Therefore if our domain is all real numbers superior to 0 (where the limit 0<a<1 and 1<b), we have to do 2 integrals:
A=A(a<x<1)+A(1<x<b)
[tex]A(a<x<1)=\int\limits^1_a {\int\limits^{x^{\frac{1}{n}} }_{x} } {} \, dy } \, dx = \int\limits^1_a {x^{\frac{1}{n} } -x \, dx = a-1 +\frac{n}{n+1} - \frac{na^{\frac{n+1}{n} } }{n+1}[/tex]
[tex]A(1<x<b)=\int\limits^b_1 {\int\limits^{x}_{x^{\frac{1}{n} } } {} \, dy } \, dx = \int\limits^b_1 {x-x^{\frac{1}{n} } \, dx =b-1 + \frac{n}{n+1} - \frac{nb^{\frac{n+1}{n} } }{n+1}[/tex]
[tex]A=a-1 +\frac{n}{n+1} - \frac{na^{\frac{n+1}{n} } }{n+1} + b-1 + \frac{n}{n+1} - \frac{nb^{\frac{n+1}{n} } }{n+1} = a+b+ \frac{2n}{n+1} - \frac{n(a^{\frac{n+1}{n} }+b^{\frac{n+1}{n} }) }{n+1} -2[/tex]
A researcher is conducting a chi-square test for independence to evaluate the relationship between gender and preference for three different designs for a new automobile. Each individual in a sample of n=30 males and n=30 females selects a favorite design from the three choices. If the researcher obtains a chi-square statistic of X^2 = 4.81, what is the appropriate statistical decision for the test?
O Fail to reject the null hypothesis with either α = 0.05 or α = 0.01
O There is not enough information to determine the appropriate decision.
O Reject the null hypothesis with α = 0.05 but not with α = 0. 01
O Reject the null hypothesis with either α = 0.05 or α = 0.01
Answer:
Step-by-step explanation:
You are assigned to the jury of a paternity case; determining whether the the child’s guardian father is actually his biological father. After listening to all the witnesses regarding the child’s family, you are 75% convinced that the guardian father is the child’s biological father. Additionally, you have been presented with laboratory blood tests indicating that the child has blood type B. The laboratory further provided population statistics stating that(a) If the guardian father is assumed to be the biological father, the child has 50% chance of having blood type B.(b) If the guardian father is assumed to NOT be the biological father, the child has 0.91% chance to have a blood type OTHER than B.How confident are you(what is the probability) that the guarding father is the child’s biological father
Answer:
Answer: 0.6022
Consider the following calculation
Step-by-step explanation:
Let F shows the event that guardian father is biological father. So
P(F) = 0.75
By the complement rule,
P(F') = 1 - P(F) =1 - 0.75 = 0.25
Let B shows the event that child has blood type B. So we have
P(B|F) = 0.50, P(B' |F') = 0.0091
By the complement rule we have
P(B|F') = 1 - P(B' |F') = 0.9909
The probability that the guarding father is the child’s biological father given that child have blood type B is
P(BFPF) P(F|B) = PRI P(BF)P(F) + P(BF)P(F) 0.50 -0.75 0.50 -0.75 +0.9909 - 0.25
0.375 0.622725 = 0.6022
Answer: 0.6022
A manufacturer of potato chips would like to know whether its bag filling machine works correctly at the 444.0 gram setting. It is believed that the machine is underfilling the bags. A 40 bag sample had a mean of 443.0 grams. A level of significance of 0.02 will be used. Determine the decision rule. Assume the standard deviation is known to be 23.0.
Answer:
We conclude that the bag filling machine works correctly at the 444.0 gram setting.
Step-by-step explanation:
We are given the following in the question:
Population mean, μ = 444.0 gram
Sample mean, [tex]\bar{x}[/tex] = 443.0 grams
Sample size, n = 40
Alpha, α = 0.02
Population standard deviation, σ = 23.0 grams
First, we design the null and the alternate hypothesis
[tex]H_{0}: \mu = 444.0\text{ grams}\\H_A: \mu < 444.0\text{ grams}[/tex]
We use one-tailed(left) z test to perform this hypothesis.
Formula:
[tex]z_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }[/tex]
Putting all the values, we have
[tex]z_{stat} = \displaystyle\frac{443 - 444}{\frac{23}{\sqrt{40}} } =-0.274[/tex]
Now, [tex]z_{critical} \text{ at 0.02 level of significance } = -2.054[/tex]
Since,
[tex]z_{stat} < z_{critical}[/tex]
We fail to reject the null hypothesis and accept the null hypothesis. Thus, we conclude that the bag filling machine works correctly at the 444.0 gram setting.
The decision rule for the hypothesis test to check machine calibration involves finding the z-value for a sample mean of 443.0 grams and comparing it with the critical z-value corresponding to a 0.02 significance level for a left-tailed test. If the z-value is less than the critical value, the null hypothesis is rejected.
Explanation:To determine the decision rule for testing whether the bag filling machine is correctly set at 444.0 grams when we suspect underfilling, we need to perform a hypothesis test using the z-test since the standard deviation is known. Given the sample size (n = 40), sample mean (μ = 443.0 grams), population mean (μ0 = 444.0 grams), population standard deviation (σ = 23.0 grams), and a level of significance of 0.02, we can formulate the null hypothesis (H0: μ = μ0) and the alternative hypothesis (H1: μ < μ0).
The decision rule involves comparing the computed z-value to the critical value from the standard normal distribution at the 0.02 level of significance for a left-tailed test. If the computed z-value is less than the critical z-value, we reject the null hypothesis and accept that the machine is underfilling. The critical z-value is found using the z-table, which corresponds to a cumulative probability of 0.02.
Please help. I do not understand how to solve this
Answer:
X=2
Step-by-step explain
4\div 10(12-3x)=3\div 10(12x-16)
when we do cross multiply then we get
40(12-3x)=30(12x-16)
we can divide by 10 both side and solve then we get
48-12x=36x-48
and after solving this equation we get
x=2
When individuals in a sample of 150 were asked whether or notthey supported capital punishment, the following information wasobtained.
Doyousupport Numberof
capitalpunishment? individuals
Yes 40
No 60
No Opinion 50
We are interested in determining whether or not the opinionsof the individuals (as to Yes, No and No Opinion) are uniformlydistributed.
The expected frequency for each group is?
a. 0.333
b. 0.50
c. 1/3
d. 50
Answer:
They are not uniformly distributed.
The expected frequency of each group is 50
Step-by-step explanation:
In probability distributions, uniform distribution refers to a probability distribution for which all of the values that a random variable can take on occur with equal probability.
In other words, for n number of events, the probability of occurrence 1,2,3,4......n is 1/n
There are 3 possible occurrence in the question above
1. Yes
2. No
3. No Opinion.
For the above events to have a uniform distribution, then they must have a probability of ⅓ each.
The expected frequency of each would then be ⅓ of n where n = 150
⅓ of 150 = 50
Sophia buys a certain brand of cereal that costs $5 per box. Yani changes to a super-saving brand of the same size. The equation shows the price, y, as a function of the number of boxes, x, for the new brand.
y = 4.35x
Part A: How many more dollars is the price of a box Sophia's original brand of cereal than the price of a box of the super-saving cereal? Show your work.
Part B: How much money does she save each month with the change in cereal brand if he buys 5 cereal boxes each month? Show your work.
$ 0.65 more dollars is the price of a box Sophia's original brand of cereal than the price of a box of the super-saving cereal
Amount saved each month with the change in cereal brand if he buys 5 cereal boxes each month is $ 3.25
Solution:
Given that Sophia buys a certain brand of cereal that costs $5 per box
The equation shows the price, y, as a function of the number of boxes, x, for the new brand:
y = 4.35x
Part A:
New brand, y = 4.35x where y is the price and x is the number of boxes
Original brand, y = 5x since given that cereal that costs $5 per box
If Sophia old cereal preference was $5, and the equation shows that the new cereal preference is $4.35, if I subtract the amount of the new one from the old,
we get , 5 - 4.35 = 0.65
Therefore, $ 0.65 more dollars is the price of a box Sophia's original brand of cereal than the price of a box of the super-saving cereal
Part B:
Given that if he buys 5 cereal boxes, let us calculate price for old and new brand
New brand, y = 4.35x
New brand, y = 4.35(5) = 21.75
Original brand, y = 5x = 5(5) = 25
Amount saved = $ 25 - $ 21.75 = $ 3.25
Thus amount saved each month with the change in cereal brand if he buys 5 cereal boxes each month is $ 3.25
Assume that foot lengths of women are normally distributed with a mean of 9.6 in and a standard deviation of 0.5 in.a. Find the probability that a randomly selected woman has a foot length less than 10.0 in.b. Find the probability that a randomly selected woman has a foot length between 8.0 in and 10.0 in.c. Find the probability that 25 women have foot lengths with a mean greater than 9.8 in.
Answer:
a) 78.81% probability that a randomly selected woman has a foot length less than 10.0 in.
b) 78.74% probability that a randomly selected woman has a foot length between 8.0 in and 10.0 in.
c) 2.28% probability that 25 women have foot lengths with a mean greater than 9.8 in.
Step-by-step explanation:
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 9.6, \sigma = 0.5[/tex].
a. Find the probability that a randomly selected woman has a foot length less than 10.0 in
This probability is the pvalue of Z when [tex]X = 10[/tex].
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{10 - 9.6}{0.5}[/tex]
[tex]Z = 0.8[/tex]
[tex]Z = 0.8[/tex] has a pvalue of 0.7881.
So there is a 78.81% probability that a randomly selected woman has a foot length less than 10.0 in.
b. Find the probability that a randomly selected woman has a foot length between 8.0 in and 10.0 in.
This is the pvalue of Z when X = 10 subtracted by the pvalue of Z when X = 8.
When X = 10, Z has a pvalue of 0.7881.
For X = 8:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{8 - 9.6}{0.5}[/tex]
[tex]Z = -3.2[/tex]
[tex]Z = -3.2[/tex] has a pvalue of 0.0007.
So there is a 0.7881 - 0.0007 = 0.7874 = 78.74% probability that a randomly selected woman has a foot length between 8.0 in and 10.0 in.
c. Find the probability that 25 women have foot lengths with a mean greater than 9.8 in.
Now we have [tex]n = 25, s = \frac{0.5}{\sqrt{25}} = 0.1[/tex].
This probability is 1 subtracted by the pvalue of Z when [tex]X = 9.8[/tex]. So:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{9.8 - 9.6}{0.1}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] has a pvalue of 0.9772.
There is a 1-0.9772 = 0.0228 = 2.28% probability that 25 women have foot lengths with a mean greater than 9.8 in.
Assume that you have a sample of n 1 equals 8n1=8, with the sample mean Upper X overbar 1 equals 42X1=42, and a sample standard deviation of Upper S 1 equals 4S1=4, and you have an independent sample of n 2 equals 15n2=15 from another population with a sample mean of Upper X overbar 2 equals 34X2=34 and a sample standard deviation of Upper S 2 equals 5S2=5. What assumptions about the two populations are necessary in order to perform thepooled-variance t test for the hypothesis Upper H 0 : μ 1 equals μ 2H0: μ1=μ2 against the alternative Upper H 1 : μ 1 >μ 2H1: μ1>μ2 and make a statistical decision?
Answer:
Check the explanation below
Step-by-step explanation:
Hello!
To make a pooled variance t-test you have to make the following assumptions:
The study variables X₁ and X₂ must be independent.
Both variables should have a normal distribution, X₁~N(μ₁; σ₁²) and X₂~N(μ₂; σ₂²)
The population variances should be equal but unknown, σ₁² = σ₂² = ?.
You have the information of two samples:
Sample 1
n₁=8
sample mean X[bar]₁= 42
sample standard deviation S₁=4
Sample 2
n₂=15
sample mean X[bar]₂= 34
sample standard deviation S₂= 5
For the hypothesis:
H₀: μ₁ = μ₂
H₁: μ₁ > μ₂
The statistic is:
t= (X[bar]₁ - X[bar]₂) - (μ₁ - μ₂) ~[tex]t_{n_1 + n_2 - 2}[/tex]
Sa[tex]\sqrt{\frac{1}{n_1} + \frac{1}{n_2} }[/tex]
Sa²= [tex]\frac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2}[/tex]
Sa²= 22
Sa= 4.69
[tex]t_{H0}[/tex]= 3.8962 ≅ 3.9
The critical region is one-tailed, for example for α: 0.05
[tex]t_{n_1 + n_2 - 2; 1 - \alpha } = t_{21; 0.95} = 1.721[/tex]
Since [tex]t_{H0}[/tex] > 1.721, then the decision is to reject the null hypothesis.
I hope it helps!
Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation dPdt=cln(KP)P where c is a constant and K is the carrying capacity.(a) Solve this differential equation for c=0.1, K=2000, and initial population P0=500. P(t)= .(b) Compute the limiting value of the size of the population. limt→[infinity]P(t)= .(c) At what value of P does P grow fastest? P= .
Answer:
A) [tex]P(t)=\frac{2000}{e^{ln4e^{-0.1t}}}[/tex]
B) P(t→∞)=2000
C) [tex]P=\frac{K}{e}=\frac{carrying capacity}{e}[/tex]
Step-by-step explanation:
Given differential eq is
[tex]\frac{dP}{dt}=c ln (\frac{K}{P})P[/tex] --- (1)
Eq is separable
[tex]\frac{1}{ln (\frac{K}{P})P}dP=cdt[/tex] --- (2)
[tex]let \\u = ln\frac{K}{P}\\du= \frac{1}{\frac{K}{P}}(\frac{-K}{P^{2}}).dP\\du=\frac{-1}{P}.dP\\dP=-P.du[/tex]
substituting in (2)
[tex]-\frac{du}{u}=dt[/tex]
Integrating both sides
[tex]\int {-\frac{1}{u}} \, du=\int{c}\,dt\\-ln|u|=ct +B\\ln|u|=-ct -B\\[/tex]
Back substituting value of u
[tex]ln |ln\frac{K}{P}|=-ct-B\\|ln\frac{K}{P}|=e^{-ct-B}\\ln|\frac{K}{P}|=be^{-ct}\\[/tex]---(3)
at t =0
[tex]ln|\frac{K}{P}|=be^{-ct}\\b=ln|\frac{K}{P}|\\b=ln\frac{2000}{500}\\b=ln|4|[/tex]
from (3)
[tex]ln|\frac{K}{P}|=be^{-ct}\\\frac{K}{P}=e^{ln4e^{-ct}}\\P(t)=\frac{K}{e^{ln4e^{-ct}}}[/tex]
[tex]P(t)=\frac{2000}{e^{ln4e^{-0.1t}}}[/tex]
B) [tex]\lim{t \to \infty}[/tex]
[tex]P( {t \to \infty} )=\frac{2000}{e^{ln4e^{-0.1\infty}}}\\e^{-0.1\infty}=0\\\implies P( {t \to \infty} )=\frac{2000}{e^{0}}}\\\\P(\infty)=2000\\[/tex]
which is the carrying capacity.
C) To find the fastest growth rate we have to maximize [tex]\frac{dP}{dt}[/tex]
From given differential eq
[tex]\frac{dP}{dt}=cln|\frac{K}{P}|P[/tex]
so function to maximize is
[tex]f(P)=cln|\frac{K}{P}|P[/tex]
[tex]f'(P)=cln|\frac{K}{P}|+c\frac{1}{\frac{K}{P}}\frac{-K}{P^{2}}.P[/tex]
[tex]f'(P)=c[ln|\frac{K}{P}|-1][/tex]
To maximize find f'(P)=0
[tex]c[ln|\frac{K}{P}|-1]=0[/tex]
[tex]ln|\frac{K}{P}|=1[/tex]
[tex]\frac{K}{P}=e[/tex]
[tex]P=\frac{K}{e}=\frac{carrying capacity}{e}[/tex]
The Gompertz function models population growth considering the carrying capacity. solve the Gompertz differential equation for c=0.1, K=2000, P0=500. The carrying capacity (K) gives the limiting value of the population size (2000). The time at which the population growth is fastest can be calculated by taking the derivative of the population function, setting it to zero and solving for P.
Explanation:The Gompertz function is a model of growth that was developed to model population growth, considering the factor of carrying capacity. Instead of compound exponential growth, it models the growth as slowing down as it reaches the limit of the carrying capacity. In your particular case, solving the differential equation for the variables provided (c=0.1, K=2000, P0=500) would require the integration techniques and use of logarithmic functions.
(a): The full solution for the Gompertz function involves advanced mathematics, you should expect some intricate function in the form of P(t) = ... The particulars depend on the specifics of the integration process.
(b): The limiting value of the population size as t→infinity (limt→[infinity]P(t) will be K, which in this instance equals 2000. This is due to the concept of carrying capacity. Beyond this value, the environment/conditions can no longer support additional growth.
(c): Finding the time at which population growth is fastest involves setting the derivative of the population function to zero and solving for P. The solution P=... can be calculated using standard techniques of calculus.
Learn more about Gompertz Function/Population Growth here:https://brainly.com/question/38330679
#SPJ11
A survey found that 73% of adults have a landline at their residence (event A); 83% have a cell phone (event B). It is known that 2% of adults have neither a cell phone nor a landline. 3. What is the probability that an adult selected at random has both a landline and a cell phone? A. 0.58 B. 0.98 C. 0.6059 D. None of these Work: 4. Given an adult has a cell phone, what is the probability he does not have a landline?
A. 0.27
B. 0.25
C. 0.3012
D. None of these
Answer:
3. What is the probability that an adult selected at random has both a landline and a cell phone?
A. 0.58
4. Given an adult has a cell phone, what is the probability he does not have a landline?
C. 0.3012
Step-by-step explanation:
We solve this problem building the Venn's diagram of these probabilities.
I am going to say that:
A is the probability that an adult has a landline at his residence.
B is the probability that an adult has a cell phone.
C is the probability that a mean is neither of those.
We have that:
[tex]A = a + (A \cap B)[/tex]
In which a is the probability that an adult has a landline but not a cell phone and [tex]A \cap B[/tex] is the probability that an adult has both of these things.
By the same logic, we have that:
[tex]B = b + (A \cap B)[/tex]
The sum of all the subsets is 1:
[tex]a + b + (A \cap B) + C = 1[/tex]
2% of adults have neither a cell phone nor a landline.
This means that [tex]C = 0.02[/tex].
73% of adults have a landline at their residence (event A); 83% have a cell phone (event B)
So [tex]A = 0.73, B = 0.83[/tex].
What is the probability that an adult selected at random has both a landline and a cell phone?
This is [tex]A \cap B[/tex].
We have that [tex]A = 0.73[/tex]. So
[tex]A = a + (A \cap B)[/tex]
[tex]a = 0.73 - (A \cap B)[/tex]
By the same logic, we have that:
[tex]b = 0.83 - (A \cap B)[/tex].
So
[tex]a + b + (A \cap B) + C = 1[/tex]
[tex]0.73 - (A \cap B) + 0.83 - (A \cap B) + (A \cap B) + 0.02 = 1[/tex]
[tex](A \cap B) = 0.75 + 0.83 - 1 = 0.58[/tex]
So the answer for question 3 is A.
4. Given an adult has a cell phone, what is the probability he does not have a landline?
83% of the adults have a cellphone.
We have that
[tex]b = B - (A \cap B) = 0.83 - 0.58 = 0.25[/tex]
25% of those do not have a landline.
So [tex]P = \frac{0.25}{0.83} = 0.3012[/tex]
The answer for question 4 is C.
Ten experts rated a newly developed chocolate chip cookie on a scale of 1 to 50. Their ratings were:
34, 35, 41, 28, 26, 29, 32, 36, 38, and 40.
1. What is the mean deviation of the ratings?
Select one:
a. 8.00
b. 4.12
c. 12.67
d. 0.75
Answer:
Option B.
Step-by-step explanation:
The given data set is
34, 35, 41, 28, 26, 29, 32, 36, 38, 40
We need to find the mean deviation of the given data.
Number of observations, n = 10
Mean of the data is
[tex]Mean=\dfrac{\sum x}{n}[/tex]
[tex]Mean=\dfrac{34+35+41+28+26+29+32+36+38+40}{10}[/tex]
[tex]Mean=\dfrac{339}{10}[/tex]
[tex]Mean=33.9[/tex]
Formula for mean deviation is
[tex]\text{Mean deviation}=\dfrac{\sum |x-mean|}{n}[/tex]
[tex]\sum |x-mean|=|34-33.9|+|35-33.9|+|41-33.9|+|28-33.9|+|26-33.9|+|29-33.9|+ |32-33.9|+|36-33.9|+|38-33.9|+|40-33.9|=41.2[/tex]
[tex]\text{Mean deviation}=\dfrac{41.2}{10}[/tex]
[tex]\text{Mean deviation}=4.12[/tex]
The mean deviation of the ratings is 4.12.
Therefore, the correct option is B.
Answer:
b. 4.12
Step-by-step explanation:
We have been given that 10 experts rated a newly developed chocolate chip cookie on a scale of 1 to 50. Their ratings were:
34, 35, 41, 28, 26, 29, 32, 36, 38, and 40.
First of all, we will find the mean of the ratings.
[tex]\text{Mean of ratings}=\frac{34+35+41+28+26+29+32+36+38+40}{10}[/tex]
[tex]\text{Mean of ratings}=\frac{339}{10}[/tex]
[tex]\text{Mean of ratings}=33.9[/tex]
Let us find absolute deviation of each point from mean.
[tex]|34-33.9|=0.1[/tex]
[tex]|35-33.9|=1.1[/tex]
[tex]|41-33.9|=7.1[/tex]
[tex]|28-33.9|=5.9[/tex]
[tex]|26-33.9|=7.9[/tex]
[tex]|29-33.9|=4.9[/tex]
[tex]|32-33.9|=1.9[/tex]
[tex]|36-33.9|=2.1[/tex]
[tex]|38-33.9|=4.1[/tex]
[tex]|40-33.9|=6.1[/tex]
Now we will use mean deviation formula.
[tex]\text{Absolute mean deviation}=\frac{\Sigma |x-\mu|}{N}[/tex], where,
[tex]\mu=\text{Mean}[/tex] and N = Number of data points.
[tex]MD=\frac{0.1+1.1+7.1+5.9+7.9+4.9+1.9+2.1+4.1+6.1}{10}[/tex]
[tex]MD=\frac{41.2}{10}[/tex]
[tex]MD=4.12[/tex]
Therefore, the mean deviation of the ratings is 4.12 and option 'b' is the correct choice.
Concerns about climate change and CO2 reduction have initiated the commercial production of blends of biodiesel (e.g., from renewable sources) and petrodiesel (from fossil fuel). Random blended fuel samples of size 35 are tested in a lab to ascertain the bio/total carbon ratio (X). If the true (i.e. population) mean is 0.948 and the true (i.e. population) standard deviation is 0.006, what is the distribution of Xbar?
Answer:
[tex]\bar X \sim N(\mu=0.948, \sigma_{\bar X}=\frac{0.006}{\sqrt{35}}=0.00101)[/tex]
Step-by-step explanation:
Previous concepts
The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
Let X the random variable who represents the bio/total carbon ratio. We know from the problem that the distribution for the parameters for the random variable X are:
[tex]\mu = 0.948[/tex]
[tex]\sigma=0.006[/tex]
We select a sample of n=35 nails. That represent the sample size.
From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
So on this case :
[tex]\bar X \sim N(\mu=0.948, \sigma_{\bar X}=\frac{0.006}{\sqrt{35}}=0.00101)[/tex]
Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. cos 41π 12.
Answer: [tex]\frac{\sqrt{6}-\sqrt{2}}{2} [/tex]
Step-by-step explanation:
We apply the formula [tex]\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y) [/tex].
Note that [tex]\cos(\frac{41}{12}\pi)=\cos((\frac{36}{12}+\frac{7}{12})\pi)=\cos(3\pi + \frac{7}{12})\pi) [/tex]. Take [tex]x=3\pi[/tex] and [tex]y=\frac{7}{12}\pi[/tex] in the formula above to get
[tex]\cos(\frac{41}{12}\pi)=\cos(3\pi)\cos(\frac{7}{12}\pi)-\sin(3\pi)\sin(\frac{7}{12}\pi)=(-1)\cdot \cos(\frac{7}{12}\pi)-0\cdot\sin(\frac{7}{12}\pi)=-\cos(\frac{7}{12}\pi)[/tex]
Then the value of this expression is [tex]-\cos(\frac{7}{12}\pi) [/tex]
We can use the cosine addition formula again to simplify further. Decompose the fraction in the argument as:
[tex]\cos(\frac{7}{12}\pi)=\cos((\frac{3}{12}+\frac{4}{12})\pi)=\cos((\frac{1}{4}\pi + \frac{1}{3})\pi) [/tex]
Applying the formula with [tex]x=\frac{1}{4}\pi[/tex] and [tex]y=\frac{1}{3}\pi[/tex] we obtain
[tex]\cos(\frac{7}{12}\pi)=\cos(\frac{1}{4}\pi)\cos(\frac{1}{3}\pi)-\sin(\frac{1}{4}\pi)\sin(\frac{1}{3}\pi)=\frac{\sqrt{2}}{2}\cdot\frac{1}{2} -\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}=\frac{\sqrt{2}-\sqrt{6}}{2} [/tex]
We conclude that this expression has the value [tex]-\frac{\sqrt{2}-\sqrt{6}}{2}=\frac{\sqrt{6}-\sqrt{2}}{2} [/tex]
Which of the following is the upper critical value of z (z*) for an 80% confidence interval?
a. 1.96
b. .84
c. 2.33
d. 1.45
e. 1.28
Answer:
e. 1.28
Step-by-step explanation:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1-0.8}{2} = 0.1[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex]. This is our critical value.
So it is z with a pvalue of [tex]1-0.1 = 0.9[/tex], so [tex]z = 1.28[/tex]
The correct answer is:
e. 1.28
Nathan and Carl are running for the mayor of Middletown, in which 60% of the voters favor Nathan and 40% support Carl. A poll is conducted in which 100 residents, selected at random, are asked their preference. What is the likelihood that the poll will show a majority in favor of Carl?
A. 0.3409
B. 0.0068
C. 0.0207
D. 0.1976
Answer:
C. 0.0207
Step-by-step explanation:
For each person, there are only two possible outcomes. Either they vote for Carl, or they vote for Nathan. So we use the binomial probability distribution.
However, we are working with samples that are considerably big. So i am going to aproximate this binomial distribution to the normal.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
In this problem, we have that:
[tex]\mu = 100*0.4 = 40[/tex]
[tex]\sigma = \sqrt{100*0.4*0.6} = 4.9[/tex]
What is the likelihood that the poll will show a majority in favor of Carl?
This is 1 subtracted by the pvalue of Z when X = 50. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{50 - 40}{4.9}[/tex]
[tex]Z = 2.04[/tex]
[tex]Z = 2.04[/tex] has a pvalue of 0.9793. So the answer is 1-0.9793 = 0.0207.
A restaurant owner is interested in the proportion of his customers who order dessert. He looks at 65 randomly selected receipts. Match the vocabulary word with its corresponding example.
The proportion of the 65 randomly selected customers who ordered dessert.
The list of the 65 Yes or No answers for whether each customer ordered dessert.
The 65 restaurant patrons whose receipts were observed by the owner.
The proportion of all customers who order dessert.
The answer: Yes or No to whether a customer ordered dessert.
All customers who come to the restaurant.
a. Data
b. Sample
c. Population
d. Statistic
e. Parameter
f. Variable
The context of a restaurant owner examining dessert orders involves several statistical concepts: Data is the list of Yes or No answers, the Sample is the 65 restaurant patrons, Population is all restaurant customers, the Statistic is the proportion of the sample who ordered dessert, the Parameter is the proportion of all customers who order dessert, and the Variable is whether an individual customer ordered dessert.
Explanation:The question involves the concepts of statistics in Mathematics, particularly focusing on how different terminologies are used. Here is the matching:
a. Data: This is the list of the 65 Yes or No answers for whether each customer ordered dessert.b. Sample: This applies to the 65 restaurant patrons whose receipts were observed by the owner.c. Population: 'All customers who come to the restaurant' is the complete set, or population in statistical terms.d. Statistic: The proportion of the 65 randomly selected customers who ordered dessert is a statistic because it provides an estimate of a particular characteristic, obtained from the sample.e. Parameter: The proportion of all customers who order dessert is the parameter. It's a numerical characteristic of the population.f. Variable: The answer: Yes or No to whether a customer ordered dessert is the variable in this context, because it can change for each customer or change over time.Learn more about Statistics Vocabulary here:https://brainly.com/question/33544588
#SPJ12
According to Inc, 79% of job seekers used social media in their job search in 2018. Many believe this number is inflated by the proportion of 22- to 30-year-old job seekers who use social media in their job search. Suppose a survey of 22- to 30-year-old job seekers showed that 314 of the 370 respondents use social media in their job search. In addition, 281 of the 370 respondents indicated they have electronically submitted a resume to an employer. (a) Conduct a hypothesis test to determine if the results of the survey justify concluding the proportion of 22- to 30-year-old job seekers who use social media in their job search exceeds the proportion of the population that use social media in their job search. Use α = 0.05. State the null and alternative hypothesis. (Enter != for ≠ as needed.)
Answer:
Null hypothesis:[tex]p\leq 0.79[/tex]
Alternative hypothesis:[tex]p > 0.79[/tex]
[tex]z=\frac{0.849 -0.79}{\sqrt{\frac{0.79(1-0.79)}{370}}}=2.786[/tex]
[tex]p_v =P(z>2.786)=0.00267[/tex]
So the p value obtained was a very low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the true proportion is higher than 0.79.
Step-by-step explanation:
1) Data given and notation
n=750 represent the random sample taken
X=314 represent the respondents that use social media in their job search.
[tex]\hat p=\frac{314}{370}=0.849[/tex] estimated proportion of respondents that use social media in their job search
[tex]p_o=0.79[/tex] is the value that we want to test
[tex]\alpha=0.05[/tex] represent the significance level
Confidence=95% or 0.95
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
2) Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that the true proportion is hgiher than 0.79.:
Null hypothesis:[tex]p\leq 0.79[/tex]
Alternative hypothesis:[tex]p > 0.79[/tex]
When we conduct a proportion test we need to use the z statisticc, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
3) Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.849 -0.79}{\sqrt{\frac{0.79(1-0.79)}{370}}}=2.786[/tex]
4) Statistical decision
It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.
The significance level provided [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.
Since is a right tailed test the p value would be:
[tex]p_v =P(z>2.786)=0.00267[/tex]
So the p value obtained was a very low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the true proportion is higher than 0.79.
A journalist reported that the average amount of time that a French person spends eating lunch at a restaurant is 22 minutes. Perform a hypothesis test to determine if a difference exists between the average time an American spends eating lunch when compared to a person from France. The following data represents the time, in minutes, that random French and American diners spent at lunch. Assume that the population variances are equal. Assume Population 1 is defined as French diners and Population 2 is defined as American diners. What is the test statistic for this hypothesis test?
American
21
17
17
20
25
16
20
16
French
24
18
20
28
18
29
17
Answer:
[tex]t=\frac{(\bar X_1 -\bar X_2)-(\mu_{1}-\mu_2)}{S_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}[/tex]
Where t follows a t distribution with [tex]n_1+n_2 -2[/tex] degrees of freedom and the pooled variance [tex]S^2_p[/tex] is given by this formula:
[tex]\S^2_p =\frac{(n_1-1)S^2_1 +(n_2 -1)S^2_2}{n_1 +n_2 -2}[/tex]
[tex]t=\frac{19 -22)-(0)}{4.095\sqrt{\frac{1}{8}+\frac{1}{7}}}=-1.416[/tex]
Step-by-step explanation:
Data given
American: 21,17,17,20,25,16,20,16 (Sample 1)
French: 24,18,20,28,18,29,17 (Sample 2)
When we have two independent samples from two normal distributions with equal variances we are assuming that
[tex]\sigma^2_1 =\sigma^2_2 =\sigma^2[/tex]
And the statistic is given by this formula:
[tex]t=\frac{(\bar X_1 -\bar X_2)-(\mu_{1}-\mu_2)}{S_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}[/tex]
Where t follows a t distribution with [tex]n_1+n_2 -2[/tex] degrees of freedom and the pooled variance [tex]S^2_p[/tex] is given by this formula:
[tex]S^2_p =\frac{(n_1-1)S^2_1 +(n_2 -1)S^2_2}{n_1 +n_2 -2}[/tex]
This last one is an unbiased estimator of the common variance [tex]\sigma^2[/tex]
The system of hypothesis on this case are:
Null hypothesis: [tex]\mu_1 = \mu_2[/tex]
Alternative hypothesis: [tex]\mu_1 \neq \mu_2[/tex]
Or equivalently:
Null hypothesis: [tex]\mu_1 - \mu_2 = 0[/tex]
Alternative hypothesis: [tex]\mu_1 -\mu_2 \neq 0[/tex]
Our notation on this case :
[tex]n_1 =8[/tex] represent the sample size for group 1
[tex]n_2 =7[/tex] represent the sample size for group 2
[tex]\bar X_1 =19[/tex] represent the sample mean for the group 1
[tex]\bar X_2 =22[/tex] represent the sample mean for the group 2
[tex]s_1=3.117[/tex] represent the sample standard deviation for group 1
[tex]s_2=5.0[/tex] represent the sample standard deviation for group 2
First we can begin finding the pooled variance:
[tex]S^2_p =\frac{(8-1)(3.117)^2 +(7 -1)(5.0)^2}{8 +7 -2}=16.770[/tex]
And the deviation would be just the square root of the variance:
[tex]S_p=4.095[/tex]
And now we can calculate the statistic:
[tex]t=\frac{19 -22)-(0)}{4.095\sqrt{\frac{1}{8}+\frac{1}{7}}}=-1.416[/tex]
Now we can calculate the degrees of freedom given by:
[tex]df=8+7-2=13[/tex]
And now we can calculate the p value using the altenative hypothesis:
[tex]p_v =2*P(t_{13}<-1.416) =0.1803[/tex]
So with the p value obtained and using the significance level assumed [tex]\alpha=0.1[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 10% of significance we don't have significant differences between the two means.
Suppose a 95% confidence interval for the average amount of weight loss on a diet program for males is between 13.0 and 18.0 pounds. These results were based on a sample of 42 male participants who were deemed to be overweight at the start of the 4-month study. What is the margin of error for this study?
Answer:
The margin of error for this study is 2.5 pounds.
Step-by-step explanation:
The margin of error is the subtraction of the mean by the lower end of the confidence interval, and this must be equal to the subtraction of the upper end to the mean.
In this problem, we have that:
M - 13 = 18 - M
2M = 31
M = 15.5
The mean is 15.5 pounds.
So the margin of error for this study is 15.5 - 13 = 18 - 15.5 = 2.5 pounds.
The test statistic of z equals=2.17 is obtained when testing the claim that pnot equals≠0.2170.217. a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed. b. Find the P-value. c. Using a significance level of alphaαequals=0.010.01, should we reject Upper H 0H0 or should we fail to reject Upper H 0H0?
Answer:
a) Two tailed test
Null hypothesis:[tex]p=0.217[/tex]
Alternative hypothesis:[tex]p \neq 0.217[/tex]
b) [tex]p_v =2*P(Z>2.17)=0.03[/tex]
c) If we compare the p value obtained and the significance level given [tex]\alpha=0.01[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis.
We Fail to reject the null hypothesis H0
Step-by-step explanation:
Data given and notation
n represent the random sample taken
X represent the outcomes desired in the sample
[tex]\hat p[/tex] estimated proportion of interest
[tex]p_o[/tex] is the value that we want to test
[tex]\alpha=0.01[/tex] represent the significance level
Confidence=99% or 0.99
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that the proportion is 0.217 or no:
a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.
Two tailed test
Null hypothesis:[tex]p=0.217[/tex]
Alternative hypothesis:[tex]p \neq 0.217[/tex]
When we conduct a proportion test we need to use the z statisitc, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
Calculate the statistic
For this case the calculated value is given z =2.17
Statistical decision
It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.
b. Find the P-value
The significance level provided [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.
Since is a bilateral test the p value would be:
[tex]p_v =2*P(Z>2.17)=0.03[/tex]
c. Using a significance level of alphaαequals=0.01, should we reject Upper H 0 or should we fail to reject Upper H 0?
If we compare the p value obtained and the significance level given [tex]\alpha=0.01[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis.
We Fail to reject the null hypothesis H0
Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable.
(a)The number of points scored during a basketball game.
(b) The weight of a T-bone steak.
(A) Is the number of points scored during a basketball game discrete or continuous?
a.The random variable is continuous. The possible value are x=0,1,2,
b.The random variable is discrete. The possible values are x ? 0
Answer:
a) Discrete
b) continuous
Step-by-step explanation:
Given that there are two random variables
1) The number of points scored during a basketball game
2) The weight of a
T bone steak.
The number of points scored during a basketball game
-- Can take values as 0,1,2.....
This can be counted i.e. we can have a one to one correspondence with natural numbers.
So this is a discrete variable
The weight of a T-bone steak
-- This can take any value in decimal or fraction. This can be between an interval comprising all values over the interval. Hence we cannot set one to one correspondence with set of natural numbers.
So continuous variable.
The test statistic of z=2.48 is obtained when testing the claim that p > 0.7.
(a) Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.
(b) Find the P-value.
(c) Using a significance level of alpha=.05, should we reject H0 or should we fail to reject H0?
Answer:
a) Right-tailed test
b) 0.006569
c) We reject [tex]H_0[/tex].
Step-by-step explanation:
We are given the following information in the question:
[tex]z_{\text{statistic}} = 2.48[/tex]
We are testing the claim that p > 0.7
Alpha, α = 0.05
a) We have to identify type of test.
Since we test for claim p > 0.7, it is a one-tailed(right) test.
b) The p-value can be calculated with the help of standard normal table.
[tex]P(z>2.48) = 1 - P(z<2.48) = 0.006569[/tex]
C) Since the p-value is less than the significance level, we fail to accept the null hypothesis and reject the null hypothesis.
Thus, we reject [tex]H_0[/tex].
The hypothesis test in question is right-tailed. The p-value can be obtained from a Z-table or statistical software and is probably less than 0.01. Given a significance level of alpha=.05, we should reject H0 if the p-value is less than alpha.
Explanation:This question involves the concepts of hypothesis testing, p-value, and significance level which are typically covered in a college-level statistics course.
(a) The test is right-tailed because we are testing the claim that p > 0.7. A right-tailed test looks at whether the test statistic is greater than the hypothesized population parameter.
(b) To find the p-value, you would need to look up the value of z=2.48 in a standard normal (Z) table or use statistical software. The p-value represents the probability that the observed difference occurred by chance assuming the null hypothesis is true. The exact value would depend on your table or software, but generally, a z-score of 2.48 would yield a very small p-value, probably less than 0.01.
(c) If the p-value is less than your significance level (alpha=.05), you would reject the null hypothesis (H0). That happens because the evidence suggests that it is highly unlikely (less than a 5% chance) that the observed difference occurred by chance.
Learn more about Hypothesis Testing here:https://brainly.com/question/34171008
#SPJ11
A symbol used to name one or more parts of a whole or a set, or a location on a number line is a?
Answer:
Fraction
Step-by-step explanation:
Fraction is a symbol that represents a part of a whole. It consists of a numerator and a denominator. The numerator is the number above the fraction bar (also known as "Vinculum), while the denominator is the number below the fraction bar. The denominator is the total number of equal parts in a whole.
Examples of Fraction: [tex]\frac{1}{2}[/tex], [tex]\frac{2}{7}[/tex] and [tex]\frac{5}{8}[/tex].
In the first example, [tex]\frac{1}{2}[/tex], 1 is the numerator, while 2, is the denominator.
Additional Information
When the numerator is smaller than the denominator, the fraction is called "proper fraction." On the contrary, when the numerator is bigger than the denominator, the fraction is called "improper fraction."
An automobile dealer wants to see if there is a relationship between monthly sales and the interest rate. A random sample of 4 months was taken. The results of the sample are presented below. Monthly Sales (Y) Interest Rate (In Percent) (X) 22 9.2 20 7.6 10 10.4 45 5.3 a) Use the method of least squares to compute an estimated regression line. b) Obtain a measure of how well the estimated regression line fits the data.
Answer:
a) [tex]y=-6.254 x +75.064[/tex]
b) r =-0.932
The % of variation is given by the determination coefficient given by [tex]r^2[/tex] and on this case [tex]-0.932^2 =0.8687[/tex], so then the % of variation explained by the linear model is 86.87%.
Step-by-step explanation:
Assuming the following dataset:
Monthly Sales (Y) Interest Rate (X)
22 9.2
20 7.6
10 10.4
45 5.3
Part a
And we want a linear model on this way y=mx+b, where m represent the slope and b the intercept. In order to find the slope we have this formula:
[tex]m=\frac{S_{xy}}{S_{xx}}[/tex]
Where:
[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}[/tex]
[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}[/tex]
With these we can find the sums:
[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=278.65-\frac{32.5^2}{4}=14.5875[/tex]
[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i){n}}=696.9-\frac{32.5*97}{4}=-91.225[/tex]
And the slope would be:
[tex]m=\frac{-91.225}{14.5875}=-6.254[/tex]
Nowe we can find the means for x and y like this:
[tex]\bar x= \frac{\sum x_i}{n}=\frac{32.5}{4}=8.125[/tex]
[tex]\bar y= \frac{\sum y_i}{n}=\frac{97}{4}=24.25[/tex]
And we can find the intercept using this:
[tex]b=\bar y -m \bar x=24.25-(-6.254*8.125)=75.064[/tex]
So the line would be given by:
[tex]y=-6.254 x +75.064[/tex]
Part b
For this case we need to calculate the correlation coefficient given by:
[tex]r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}[/tex]
[tex]r=\frac{4(696.9)-(32.5)(97)}{\sqrt{[4(278.65) -(32.5)^2][4(3009) -(97)^2]}}=-0.937[/tex]
So then the correlation coefficient would be r =-0.932
The % of variation is given by the determination coefficient given by [tex]r^2[/tex] and on this case [tex]-0.932^2 =0.8687[/tex], so then the % of variation explained by the linear model is 86.87%.
Suppose that an automobile manufacturer designed a radically new lightweight engine and wants to recommend the grade of gasoline that will have the best fuel economy. The four grades are: regular, below regular, premium, and super premium. The test car made three trial runs on the test track using each of the four grades and the miles per gallon recorded. At the 0.05 level, what is the critical value of F used to test the hypothesis that the miles per gallon for each fuel is the same. Show your work.
Miles Per Gallon
Regular: Below Regular Premium Super Premium
$39.31 36.69 38.99 40.04
39.87 40.00 40.02 39.89
39.87 41.01 39.99 39.93
A.) 3.49
B.) 4.07
C.) 2.33
D.) 3.86
E.) 3.26
Answer:
B) 4.07
Step-by-step explanation:
First we need to calculate the mean of all the data, which is the same as the mean of the means of each grade of gasoline:
Regular BelowRegular Premium SuperPremium
39.31 36.69 38.99 40.04
39.87 40.00 40.02 39.89
39.87 41.01 39.99 39.93
X1⁻=39.68 X2⁻= 39.23 X3⁻= 39.66 X4⁻= 39.95
Xgrand⁻ = (39.68+39.23+39.66+39.95)/4 = 39.63
Next we need to calculate the sum of squares within the group (SSW) and the sum of squares between the groups (SSB), and the respective degrees of freedom):
SSW = [ (39.31-39.68)² + (39.87-39.68)² + (39.87-39.68)² ] + [ (36.69-39.23)² + (40.00-39.23)² + (41.01-39.23)² ] + [ (38.99-39.66)² + (40.02-39.66)² + (39.99-39.66)² ] + [ (40.04-39.95)² + (39.89-39.95)² + (39.93-39.95)² ] = [0.2091] + [10.2129] + [0.6874] + [0.0121] = 11.12
SSW = 11.12
Degrees of freedom in this case is calculated by m(n-1), with m being the number of grades of gasoline (4) and n being the number of trial results for each one (3), so we would have 4(3-1) = 8 degrees of freedom
SSB = [ (39.68-39.63)² + (39.68-39.63)² + (39.68-39.63)²] + [ (39.23-39.63)² + (39.23-39.63)² + (39.23-39.63)² ] + [ (39.66-39.63)² + (39.66-39.63)² + (39.66-39.63)² ] + [ (39.95-39.63)² + (39.95-39.63)² +(39.95-39.63)² ] = [0.0075] + [0.48] + [0.0027] + [0.3072] = 0.7974
SSB = 0.80
For this case, the degrees of freedom are m-1, so we would have 4-1 = 3 degrees of freedom
Now we can establish the hypothesis for the test:
H0: μ1 = μ2 = μ3 = μ4
The null hypothesis states that the means of miles per gallon for each fuel are the same, indicating that the drade of gasoline does not make a difference, therefore our alternative hypothesis will be:
H1: the grade of gasoline does makes a difference
We will use the F statistic to test the hypothesis, which is calculated like follows:
F - statistic = (SSB/m-1) / (SSW/m(n-1)) = (0.80/3) / (11.12/8) = 0.19
We know that the level of significance we are using is α = 0.05, so to find the critical value F we need to look at some table of critical values for the F distribution for the 0.05 significance level (like the attached image). Then we just need to look fot the value that is located in the intersection between the degrees of freedom we have in the numerator (horizontal) and the denominator (vertical) of the statistic (3 and 8). That critical value is:
Fc = 4.07
Final answer:
The critical value of F used to test the hypothesis that the miles per gallon for each fuel is the same is 3.49.
Explanation:
To test the hypothesis that the miles per gallon for each fuel is the same, we can use an ANOVA test. The critical value of F at the 0.05 level can be found using the F-distribution table or by using statistical software. Since we have three trial runs for each grade of gasoline, we have a total of 12 observations. At a significance level of 0.05 and with 3 degrees of freedom for the numerator and 8 degrees of freedom for the denominator, the critical value of F is 3.49.
By determining f prime left parenthesis x right parenthesis equals ModifyingBelow lim With h right arrow 0 StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFractionf′(x)=limh→0 f(x+h)−f(x) h, find f prime left parenthesis 7 right parenthesisf′(7) for the given function. f left parenthesis x right parenthesis equals 5 x squaredf(x)=5x2 f prime left parenthesis 7 right parenthesisf′(7)equals=nothing (Simplify your answer.)
Answer:
70 is answer
Step-by-step explanation:
Given that a function in x is
[tex]f(x) = 5x^2[/tex]
we have to find f'(7)
we know by derivative rule derivative of a function is
[tex]f'(x) = lim_({h-->0}) \frac{f(x+h)-f(x)}{h}[/tex]
For finding out at 7 we replace x by 7
[tex]f'(7) = lim_({h-->0}) \frac{f(7+h)-f(7)}{h}[/tex]
=[tex]lim\frac{5(7+h)^2-5*7^2}{h} \\= lim \frac{10h*7+h^2}{h} \\= 70+h = 70[/tex]
So f'(7) = 70
answer is 70
Answer:
f'(7)=70
Step-by-step explanation:
We have the definition of the derivative as:
[tex]f'(x)= \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}[/tex]
Now we have a function [tex]f(x)=5x^2[/tex] and we want to approximate the first derivative around x=7, that is [tex]f'(7)[/tex].
We can replace this in the first formula as:
[tex]f'(x)= \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}= \lim_{h \to 0} \dfrac{5(x+h)^2-5x^2}{h}\\\\f'(x)=\lim_{h \to 0} \dfrac{5(x^2+2xh+h^2-x^2)}{h}\\\\f'(x)=\lim_{h \to 0}\dfrac{5(2xh+h^2)}{h}\\\\f'(x)=\lim_{h \to 0}5(2x+h)\\\\f'(x)=10x+lim_{h \to 0}h=10x+0=10x[/tex]
Then, the value for f'(7) is:
[tex]f'(7)=10\cdot 7=70[/tex]
For the Sequence 2,6,10,14. Write the recursive rule, and a explicit rule
Answer:
The recursive formula for given series is: [tex]a_{n} = a_{n-1} + 4[/tex]
The explicit formula is given by: [tex]a_{n} = 10 + 5(n-1)[/tex]
Step-by-step explanation:
Determining the Recursive Formula:
As the given sequence is: 2,6,10,14The difference d can be computed by taking the difference between the consecutive terms of the mentioned sequence.d = 6 -2 = 10 - 6 = 14 - 10 = 4
It is observed that the difference between the consecutive terms remains constant) with common difference d = 4We know that if the difference between the consecutive terms remains constant), then the series is in arithmetic series. The recursive formula is: [tex]a_{n} = a_{n-1} + d[/tex]So, the recursive formula for given series is: [tex]a_{n} = a_{n-1} + 4[/tex]Determining the Explicit Rule or Formula:
An explicit formula defines the nth term of the sequence, where n being the term's location. In other words, a sequence can be defined as a formula in terms of n. So,
First determine the sequence whether the sequence is in arithmetic. As we know that 2,6,10,14 is in arithmetic. Then find the common difference. Here, d = 6 -2 = 10 - 6 = 14 - 10 = 4Establishing an explicit formula by analyzing the pattern. i.e. adding first term to the product of d (common difference) and one less than the term numberHence, the explicit formula is given by:
[tex]a_{n} = a_{1} + d (n-1)[/tex]where,
aₙ is the nth term of the sequence
n is the term number
a₁ is the first term
d is the common difference
As the given sequence is: 2,6,10,14
a₁ = first term = 2
d = common difference = 6 - 2 = 4
Using explicit formula:
[tex]a_{n} = a_{1} + d (n-1)[/tex]
[tex]a_{n} = 10 + 5(n-1)[/tex]
[tex]a_{n} = 10 + 5n-5[/tex]
[tex]a_{n} = 5n+ 5[/tex]
Keywords: recursive formula, explicit formula, sequence
Learn more about recursive formula brainly.com/question/11673288
#learnwithBrainly
Final answer :
The recursive rule for the sequence is Sn = Sn- 1 4 with S1 = 2, and the unequivocal rule is Sn = 4n- 2. Both rules help in determining the sequence's terms either by using the former term or directly calculating any term's value.
Explanation:
To determine the recursive rule and the unequivocal rule for the sequence 2, 6, 10, 14, we need to observe the pattern of the sequence and produce formulas that can induce the terms of the sequence. This sequence is computation because the difference between each term is constant.
Recursive Rule
A recursive rule for a sequence defines each term grounded on the former term. For the given sequence, each term after the first is attained by adding 4 to the former term. The recursive rule can be written as Sn = Sn- 1 4, for n> 1 where S1 = 2 is the first term.
unequivocal Rule
An unequivocal rule, also known as a unrestricted- form expression, defines the utmost term of a sequence as a function of n without demanding the former term. For this computation sequence, the unequivocal rule can be determined by using the formula Sn = a( n- 1) d, where' a' is the first term and'd' is the common difference. The unequivocal rule for our sequence is Sn = 2( n- 1) × 4 = 4n- 2.