CED 6030, section 02, Fall 2015 Test 2 (Exam)

| August 30, 2017

Question
11/13/2015 Test 2
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Due : Friday, November 13 2015 08:00 PM EST
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(a) If we have a distribution of x values that is more or less mound­shaped and somewhat symmetrical, what is the sample size n
needed to claim that the distribution of sample means x from random samples of that size is approximately normal?
n ?
(b) If the original distribution of x values is known to be normal, do we need to make any restriction about sample size in order to claim
that the distribution of sample means x taken from random samples of a given size is normal?
Yes
No
1. Question DetailsBBUnderStat11 6.5.007.
Test 2 (Exam)

CED 6030, section 02, Fall 2015
Instructor: He Wang
WebAssign
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Briefly answer the following questions.
(a) What is a null hypothesis H0?
A specific hypothesis where the claim is that the population parameter does not equal 0.
A specific hypothesis where the claim is that the population parameter is equal to 0.
Any hypothesis that differs from the original claim being made.
A working hypothesis making a claim about the population parameter in question.
(b) What is an alternate hypothesis H1?
A specific hypothesis where the claim is that the population parameter does not equal 0.
Any hypothesis that differs from the original claim being made.
A specific hypothesis where the claim is that the population parameter is equal to 0.
A working hypothesis making a claim about the population parameter in question.
(c) What is a type I error?
Type I error is rejecting the null hypothesis when it is false.
Type I error is failing to reject the null hypothesis when it is false.
Type I error is rejecting the null hypothesis when it is true.
Type I error is failing to reject the null hypothesis when it is true.
What is a type II error?
Type II error is rejecting the null hypothesis when it is false.
Type II error is failing to reject the null hypothesis when it is false.
Type II error is rejecting the null hypothesis when it is true.
Type II error is failing to reject the null hypothesis when it is true.
(d) What is the level of significance of a test?
The probability of a type II error.
The probability of a type I error.
What is the probability of a type II error?
1 ? ?
?
?
1 ? ?
2. Question DetailsBBUnderStat11 8.1.001.
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Gentle Ben is a Morgan horse at a Colorado dude ranch. Over the past 8 weeks, a veterinarian took the following glucose readings from
this horse (in mg/100 ml).
94 86 81 105 97 109 82 90
The sample mean is x ? 93.0. Let x be a random variable representing glucose readings taken from Gentle Ben. We may assume that x
has a normal distribution, and we know from past experience that ? = 12.5. The mean glucose level for horses should be ? = 85
mg/100 ml.† Do these data indicate that Gentle Ben has an overall average glucose level higher than 85? Use ? = 0.05.
(a) What is the level of significance?
State the null and alternate hypotheses. Will you use a left­tailed, right­tailed, or two­tailed test?
H0: ? > 85? H1: ? = 85? right­tailed
H0: ? = 85? H1: ? 85? right­tailed
H0: ? = 85? H1: ? ? 85? two­tailed
(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.
The standard normal, since we assume that x has a normal distribution with unknown ?.
The standard normal, since we assume that x has a normal distribution with known ?.
The Student’s t, since n is large with unknown ?.
The Student’s t, since we assume that x has a normal distribution with known ?.
What is the value of the sample test statistic? (Round your answer to two decimal places.)
(c) Find (or estimate) the P­value. (Round your answer to four decimal places.)
Sketch the sampling distribution and show the area corresponding to the P­value.
3. Question DetailsBBUnderStat11 8.1.020.
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(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically
significant at level ??
At the ? = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
At the ? = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.
At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
(e) State your conclusion in the context of the application.
There is sufficient evidence at the 0.05 level to conclude that Gentle Ben’s glucose is higher than 85 mg/100 ml.
There is insufficient evidence at the 0.05 level to conclude that Gentle Ben’s glucose is higher than 85 mg/100 ml.
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How much customers buy is a direct result of how much time they spend in the store. A study of average shopping times in a large
national houseware store gave the following information (Source: Why We Buy: The Science of Shopping by P. Underhill).
Women with female companion: 8.3 min.
Women with male companion: 4.5 min.
Suppose you want to set up a statistical test to challenge the claim that a woman with a female friend spends an average of 8.3
minutes shopping in such a store.
(a) What would you use for the null and alternate hypotheses if you believe the average shopping time is less than 8.3
minutes?
Ho: ? = 8.3? H1: ? < 8.3
Ho: ? 8.3
Ho: ? = 8.3? H1: ? ? 8.3
Is this a right­tailed, left­tailed, or two­tailed test?
left­tailed
two­tailed
right­tailed
(b) What would you use for the null and alternate hypotheses if you believe the average shopping time is different from 8.3
minutes?
Ho: ? = 8.3? H1: ? > 8.3
Ho: ? = 8.3? H1: ? ? 8.3
Ho: ? ? 8.3? H1: ? = 8.3
Ho: ? = 8.3? H1: ? < 8.3
Is this a right­tailed, left­tailed, or two­tailed test?
left­tailed
right­tailed
two­tailed
Stores that sell mainly to women should figure out a way to engage the interest of men! Perhaps comfortable seats and a big TV with
sports programs. Suppose such an entertainment center was installed and you now wish to challenge the claim that a woman with a
male friend spends only 4.5 minutes shopping in a houseware store.
(c) What would you use for the null and alternate hypotheses if you believe the average shopping time is more than 4.5
minutes?
Ho: ? = 4.5? H1: ? 4.5
Ho: ? = 4.5? H1: ? ? 4.5
Ho: ? > 4.5? H1: ? = 4.5
Is this a right­tailed, left­tailed, or two­tailed test?
right­tailed
two­tailed
left­tailed
4. Question DetailsBBUnderStat11 8.1.016.
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(d) What would you use for the null and alternate hypotheses if you believe the average shopping time is different from 4.5
minutes?
Ho: ? = 4.5? H1: ? > 4.5
Ho: ? = 4.5? H1: ? ? 4.5
Ho: ? ? 4.5? H1: ? = 4.5
Ho: ? = 4.5? H1: ? 0
Ho: ?d = 0
Ho: ?d ? 0
Ho: ?d 5 and nq > 5 are satisfied in this problem? Explain why this would be an important
consideration.
Yes, the conditions are satisfied. This is important because it allows us to say that p? is approximately normal.
No, the conditions are not satisfied. This is important because it allows us to say that p? is approximately normal.
No, the conditions are not satisfied. This is important because it allows us to say that p? is approximately binomial.
Yes, the conditions are satisfied. This is important because it allows us to say that p? is approximately binomial.
8. Question DetailsBBUnderStat11 7.3.012.
If we fail to reject (i.e., “accept”) the null hypothesis, does this mean that we have proved it to be true beyond all doubt? Explain your
answer.
No, it suggests that the evidence is not sufficient to merit rejecting the null hypothesis.
Yes, it suggests that the evidence is sufficient to merit rejecting the alternative hypothesis beyond all doubt.
No, it suggests that the null hypothesis is true only some of the time.
Yes, if we fail to reject the null we have found evidence that the null is true beyond all doubt.
9. Question DetailsBBUnderStat11 8.1.003.
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(a) Suppose n = 6 and the sample correlation coefficient is r = 0.884. Is r significant at the 1% level of significance (based on a twotailed
test)? (Round your answers to three decimal places.)
t =
critical t =
Conclusion:
Yes, the correlation coefficient ? is significantly different from 0 at the 0.01 level of significance.
No, the correlation coefficient ? is not significantly different from 0 at the 0.01 level of significance.
(b) Suppose n = 10 and the sample correlation coefficient is r = 0.884. Is r significant at the 1% level of significance (based on a twotailed
test)? (Round your answers to three decimal places.)
t =
critical t =
Conclusion:
Yes, the correlation coefficient ? is significantly different from 0 at the 0.01 level of significance.
No, the correlation coefficient ? is not significantly different from 0 at the 0.01 level of significance.
(c) Explain why the test results of parts (a) and (b) are different even though the sample correlation coefficient r = 0.884 is the same
in both parts. Does it appear that sample size plays an important role in determining the significance of a correlation coefficient?
Explain.
As n decreases, the degrees of freedom and the test statistic increase. This produces a smaller P value.
As n increases, so do the degrees of freedom, and the test statistic. This produces a smaller P value.
As n increases, the degrees of freedom and the test statistic decrease. This produces a smaller P value.
As n increases, so do the degrees of freedom, and the test statistic. This produces a larger P value.
10.Question DetailsBBUnderStat11 9.3.013.
Suppose you want to eat lunch at a popular restaurant. The restaurant does not take reservations, so there is usually a waiting time
before you can be seated. Let x represent the length of time waiting to be seated. From past experience, you know that the mean
waiting time is ? = 18.6 minutes with ? = 3.5 minutes. You assume that the x distribution is approximately normal. (Round your
answers to four decimal places.)
(a) What is the probability that the waiting time will exceed 20 minutes, given that it has exceeded 15 minutes? Hint: Compute
P(x > 20|x > 15).
(b) What is the probability that the waiting time will exceed 25 minutes, given that it has exceeded 18 minutes? Hint: Compute
P(x > 25|x > 18).
11.Question DetailsBBUnderStat11 6.3.039.
Sketch the area under the standard normal curve over the indicated interval and find the specified area. (Round your answer to four
decimal places.)
The area between and z = 1.42 is .
12.Question DetailsBBUnderStat11 6.2.025.
z = ?2.28
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When are two random samples independent?
When sample data drawn from one population are completely unrelated to the selection of data from the other population.
Random samples are independent by nature of random sampling.
When sample data drawn from one population are somewhat unrelated to the selection of data from the other population.
When sample data drawn from one population are related to the selection of data from the other population.
13.Question DetailsBBUnderStat11 7.4.001.
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Socially conscious investors screen out stocks of alcohol and tobacco makers, firms with poor environmental records, and companies
with poor labor practices. Some examples of “good,” socially conscious companies are Johnson and Johnson, Dell Computers, Bank of
America, and Home Depot. The question is, are such stocks overpriced? One measure of value is the P/E, or price­to­earnings ratio.
High P/E ratios may indicate a stock is overpriced. For the S&P Stock Index of all major stocks, the mean P/E ratio is ? = 19.4. A
random sample of 36 “socially conscious” stocks gave a P/E ratio sample mean of x = 17.9, with sample standard deviation s = 5.8.
Does this indicate that the mean P/E ratio of all socially conscious stocks is different (either way) from the mean P/E ratio of the S&P
Stock Index? Use ? = 0.05.
(a) What is the level of significance?
State the null and alternate hypotheses.
H0: ? = 19.4? H1: ? ? 19.4
H0: ? = 19.4? H1: ? 19.4
H0: ? > 19.4? H1: ? = 19.4
(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.
The standard normal, since the sample size is large and ? is known.
The Student’s t, since the sample size is large and ? is known.
The standard normal, since the sample size is large and ? is unknown.
The Student’s t, since the sample size is large and ? is unknown.
What is the value of the sample test statistic? (Round your answer to three decimal places.)
(c) Find the P­value. (Round your answer to four decimal places.)
Sketch the sampling distribution and show the area corresponding to the P­value.
14.Question DetailsBBUnderStat11 8.2.015.
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(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically
significant at level ??
At the ? = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
At the ? = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.
At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
(e) Interpret your conclusion in the context of the application.
There is sufficient evidence at the 0.05 level to conclude that the mean P/E ratio of all socially conscious stocks differs
from the mean P/E ratio of the S&P Stock Index.
There is insufficient evidence at the 0.05 level to conclude that the mean P/E ratio of all socially conscious stocks differs
from the mean P/E ratio of the S&P Stock Index.
Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your
answer to four decimal places.)
? = 4.6? ? = 1.7
P(3 ? x ? 6) =
15.Question DetailsBBUnderStat11 6.3.005.
In the least­squares line = 5 – 9x, what is the value of the slope?
When x changes by 1 unit, by how much does y change?
When x decreases by 1 unit, y decreases by 9 units.
When x increases by 1 unit, y decreases by ?9 units.
When x increases by 1 unit, y decreases by 9 units.
When x increases by 1 unit, y increases by 9 units.
16.Question DetailsBBUnderStat11 9.2.001.
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Quick Start Company makes 12­volt car batteries. After many years of product testing, the company knows that the average life of a
Quick Start battery is normally distributed, with a mean of 45.0 months and a standard deviation of 8.3 months.
(a) If Quick Start guarantees a full refund on any battery that fails within the 36­month period after purchase, what percentage
of its batteries will the company expect to replace? (Round your answer to two decimal places.)
%
(b) If Quick Start does not want to make refunds for more than 11% of its batteries under the full­refund guarantee policy, for
how long should the company guarantee the batteries (to the nearest month)?
months
17.Question DetailsBBUnderStat11 6.3.029.
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Use the following linear regression equation to answer the questions.
x1 = 1.9 + 3.1×2 – 7.7×3 + 2.3×4
(a) Which variable is the response variable?
x1
x3
x2
x4
Which variables are the explanatory variables? (Select all that apply.)
x4
x3
x1
x2
(b) Which number is the constant term? List the coefficients with their corresponding explanatory variables.
constant
x2 coefficient
x3 coefficient
x4 coefficient
(c) If x2 = 6, x3 = 10, and x4 = 1, what is the predicted value for x1? (Use 1 decimal place.)
(d) Explain how each coefficient can be thought of as a “slope” under certain conditions.
If we look at all coefficients together, each one can be thought of as a “slope.”
If we look at all coefficients together, the sum of them can be thought of as the overall “slope” of the regression line.
If we hold all explanatory variables as fixed constants, the intercept can be thought of as a “slope.”
If we hold all other explanatory variables as fixed constants, then we can look at one coefficient as a “slope.”
Suppose x3 and x4 were held at fixed but arbitrary values and x2 increased by 1 unit. What would be the corresponding change
in x1?
Suppose x2 increased by 2 units. What would be the expected change in x1?
Suppose x2 decreased by 4 units. What would be the expected change in x1?
(e) Suppose that n = 9 data points were used to construct the given regression equation and that the standard error for the
coefficient of x2 is 0.314. Construct a 99% confidence interval for the coefficient of x2. (Use 2 decimal places.)
lower limit
upper limit
(f) Using the information of part (e) and level of significance 1%, test the claim that the coefficient of x2 is different from zero.
18.Question DetailsBBUnderStat11 9.4.001.
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(Use 2 decimal places.)
t
t critical ±
Conclusion
Reject the null hypothesis, there is sufficient evidence that ?2 differs from 0.
Reject the null hypothesis, there is insufficient evidence that ?2 differs from 0.
Fail to reject the null hypothesis, there is insufficient evidence that ?2 differs from 0.
Fail to reject the null hypothesis, there is sufficient evidence that ?2 differs from 0.
Explain how the conclusion of this test would affect the regression equation.
If we conclude that ?2 is not different from 0 then we would remove x1 from the model.
If we conclude that ?2 is not different from 0 then we would remove x2 from the model.
If we conclude that ?2 is not different from 0 then we would remove x4 from the model.
If we conclude that ?2 is not different from 0 then we would remove x3 from the model.
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Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as
gout, leukemia, or lymphoma.† Over a period of months, an adult male patient has taken fourteen blood tests for uric acid. The mean
concentration was x = 5.35 mg/dl. The distribution of uric acid in healthy adult males can be assumed to be normal, with
? = 1.75 mg/dl.
(a) Find a 95% confidence interval for the population mean concentration of uric acid in this patient’s blood. What is the margin
of error? (Round your answers to two decimal places.)
lower limit
upper limit
margin of error
(b) What conditions are necessary for your calculations? (Select all that apply.)
? is known
uniform distribution of uric acid
n is large
normal distribution of uric acid
? is unknown
(c) Interpret your results in the context of this problem.
There is a 5% chance that the confidence interval is one of the intervals containing the population average uric acid level
for this patient.
There is a 95% chance that the confidence interval is one of the intervals containing the population average uric acid
level for this patient.
There is not enough information to make an interpretation.
The probability that this interval contains the true average uric acid level for this patient is 0.05.
The probability that this interval contains the true average uric acid level for this patient is 0.95.
(d) Find the sample size necessary for a 95% confidence level with maximal margin of error E = 1.06 for the mean
concentration of uric acid in this patient’s blood. (Round your answer up to the nearest whole number.)
blood tests
19.Question DetailsBBUnderStat11 7.1.016.
Binomial probability distributions depend on the number of trials n of a binomial experiment and the probability of success p on each
trial. Under what conditions is it appropriate to use a normal approximation to the binomial? (Select all that apply.)
nq > 5
np > 10
p 0.5
nq > 10
np > 5
20.Question DetailsBBUnderStat11 6.6.001.
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What are the values of the mean and standard deviation of a standard normal distribution?
? =
? =
21.Question DetailsBBUnderStat11 6.2.004.
Suppose two variables are negatively correlated. Does the response variable increase or decrease as the explanatory variable
increases?
The response variable will remain constant as the explanatory variable increases.
The response variable will increase as the explanatory variable increases.
We can not say whether the response variable will increase or decrease.
The response variable will decrease as the explanatory variable increases.
22.Question DetailsBBUnderStat11 9.1.003.
Suppose the heights of 18­year­old men are approximately normally distributed, with mean 65 inches and standard deviation 5 inches.
(a) What is the probability that an 18­year­old man selected at random is between 64 and 66 inches tall? (Round your answer
to four decimal places.)
(b) If a random sample of eighteen 18­year­old men is selected, what is the probability that the mean height x is between 64
and 66 inches? (Round your answer to four decimal places.)
(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?
The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.
The probability in part (b) is much higher because the mean is smaller for the x distribution.
The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.
The probability in part (b) is much higher because the mean is larger for the x distribution.
The probability in part (b) is much higher because the standard deviation is larger for the x distribution.
23.Question DetailsBBUnderStat11 6.5.014.

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