1. Summer Wages. Suppose Iowa State is interested in the hourly wages earned by their undergraduate students over the summer. It is known that the distribution of hourly wages for all undergraduates in the United States is right skewed. The university randomly samples forty-nine Iowa State students and compiles information about their summer jobs. The sample mean was $10.25/hr and the standard deviation was $0.70/hr.
(A) Construct a 98% confidence interval for the mean hourly summer wage of Iowa State students. ROUND YOUR ANSWERS TO THE NEAREST 2 DECIMAL PLACES.
Show work here:
x¯ = $10.25, s = $0.70, n = 49, C = 98
d.f.= 48, round down to 40, t∗= 2.423 (from Table D) CI = 10.
Lower Bound: 10.01Upper Bound: 10.49
(B)Interpret the confidence interval you constructed in part (B) in the context of these data.
We are 98% confident that the true mean hourly summer wage of Iowa State students is between $10.01 and $10.49 an hour.
(C)The University of Iowa claims that their students make, on average, $10.07 per hour over the summer. Do Iowa State students have reasons to believe that, on average, their hourly summer wage is different than that of the University of Iowa students (based on 98% confidence)? Circle the correct answer and explain your choice.
YES / NO / NOT ENOUGH INFO
Explanation (similar for forms A and B):
$10.07 is within our 98% confidence interval, so based on 98% confidence we cannot claim that the average summer wages of Iowa State students are any different than those of University of Iowa.
(D)Circle all that apply. Which of the following would result in a narrower confidence interval than the one calculated in part (B)?
i.A sample mean of $10.50/hr.
ii.A sample standard deviation of $1.00/hr.
iii.A sample size of 100.
iv.A confidence level of 95%.
v.A sample mean of $10.00/hr.
(E) Suppose that you are told a 70% confidence interval for the mean hourly summer wage of Iowa State students is ($10.15, $10.35). What is the margin of error? FORM A: Range = 10.35 - 10.15 = .2, Margin of error =
Final Answer: (ROUND TO THE NEAREST 2 DECIMAL PLACES) .1 ($0.10)
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(F) Circle all that apply. If the ISU student representative were to take another sample of forty-nine undergraduate students at ISU, and calculate a 98% confidence interval, what would you expect to change from the interval in part (A)?
i.The lower and upper bound of the confidence interval.
ii.The sample mean.
iii.The estimated standard error. iv. The value of σ.
v. The value of s. vi. The critical value.
vii. The center of the confidence interval. viii. The value of µ.
2.Seedling Height. A biologist is interested in determining whether sunflower seedlings treated with an extract from Vinca minor roots resulted in a different average height of sunflower seedlings compared to the standard height of 13.8 cm. Data from a random sample of 100 sunflower seedlings treated with the extract was analyzed in JMP and the output is provided below.
(A)Choose the correct set of null and alternative hypotheses from the ones provided below:
•H0 :µ = 13.8 vs. Ha : µ < 13.8Ha : ¯x < 13.8
•H0 :µ = 13.8 vs. Ha : µ ≤ 13.8Ha : µ ≥ 13.8
•H0 :µ ≥ 13.8 vs. Ha : µ < 13.8• H : µ ≤ 13.8 vs. Ha : µ < 13.8
•H0 :µ = 13.8 vs. Ha : µ > 13.8Ha : µ 6= 13.8
•H0 :µ = 13.8 vs. Ha : µ > 13.8Ha : µ = 13.8
•H0 : ¯x = 13.8 vs. Ha : ¯x < 13.8Ha : ¯x 6= 13.8
•None of the above.
(B)Fill in the blanks: A Type I error in the context of this problem is concluding that the mean height of treated seedlings is not equal to 13.8 cm when, in fact, it is equal to 13.8cm.
(C)Based on the alternative hypothesis and using the information provided in the JMP output, what is the appropriate p-value? FORM A: 0.0376 × 2 = 0.0752 ;
(D)Make a decision at the α = 0.1 significance level. (Circle one and explain your choice.) REJECT H0 or FAIL TO REJECT H0
Explanation:
FORM A: Since the p-value=0.075≤ α = 0.1, we reject the null hypothesis.
(E)Assuming a 0.1 level of significance for the test above, calculate the corresponding 100(1−α)% confidence interval for the true mean height of sunflower seedlings treated with the extract. Show all your work. ROUND YOUR ANSWERS TO THE NEAREST 2 DECIMAL PLACES.
3.Hours of Sleep. Administrators at the Iowa State College of Business are concerned that their undergraduate students get less sleep than the recommended sleep requirement for college students, which is 6.8 hours per night. To investigate this, an advisor in the College of Business at ISU surveys a random sample of forty undergraduate business majors. Her findings and analysis are contained in the following JMP output:
(A) What is the unknown parameter that the researcher is interested in? (Choose one.)
i.The mean number of hours of sleep per night for all ISU students.
ii.The total hours of sleep per night for all ISU undergraduate Business majors.
iii.The mean number of hours of sleep per night for all ISU undergraduate Business majors.
iv.The number of hours of sleep per night of a student in the sample.
v.The mean number of hours of sleep per night for all students in the sample.
vi. The total hours of sleep per night for all ISU undergraduate Business majors.
(B)State the null and the alternative hypotheses using proper statistical notation.
FORM A :H0 : µ = 6.8versusHA : µ < 6.8
(C)Calculate the value of the test statistic for the test set-up in question (B).
Show work here: Final Answer: -2.50 (ROUND TO THE NEAREST 2 DECIMAL PLACES).
(D)Report the number degrees of freedom associated with you answer in part (C): 39;
(E)Using JMP, report here the p-value for the test in part (B) and interpret it in the context of this problem.
The correct p-value for the lower tailed alternative hypothesis is 0.0083. This value tells us that if the null hypothesis were true, we would expect to see a test statistic of -2.50 or less in approximately 0.83% of samples.
(F)Using a significance level of 0.01, write the appropriate conclusion for this test in the context of the data.
Since p = 0.0083 < α = 0.01, we have sufficient statistically significant evidence to reject the null hypothesis and conclude that the true mean average hours of sleep per night for all undergraduate business majors at Iowa State is less than 6.8.
(G)(Circle all that apply.) The statistical inference about the population mean µ was possible for which of the following reason(s)?
i.The sample size is sufficiently large to ensure the CLT (Central Limit Theorem) applies to the sampling distribution of the sample mean.
ii.The data came from a normally distributed population. iii. The sample is a random sample from the population. iv. The standard deviation of the population is known.
v. Statistical inference can always be done regardless of the underlying population distribution and of the sample size. That is, the CLT is not needed.
(H)[BONUS] Assume now that the original passage for this problem stated, ”Administrators at the Iowa State College of Business speculate that their undergraduate students get a different amount of sleep than students in other undergraduate majors at Iowa State.” Answer the following questions using the same sample of students mentioned in the statement of the problem. For each statement, circle the correct answer:
i.The test statistic to test the claim would be LESS THAN/GREATER THAN/EQUAL TO the test statistic calculated in the original problem.
ii.The p-value used to make a decision would be HALF OF/TWO TIMES/EQUAL TO the p-value selected in part (E). iii. The decision would be THE SAME AS/DIFFERENT THAN the decision in part (G).
4. TRUE or FALSE: Identify which of the following are valid statements and which ones are false.
(A)T F All other factors remaining the same, increasing the sample size will increase the precision of the confidence interval.
(B)T F All other factors remaining the same, increasing the sample size will increase the accuracy of the confidence interval.
(C)T F The mean of a population, µ, is random and hence is considered a random variable.
(D)T F We are 90% confident that a valid 90% confidence interval for the true population mean will include the sample mean.
(E)T F The p-value is the probability that the null hypothesis (H0) is true.
5. Remodeling costs. Kitchens are frequently the most expensive room for homeowners to remodel. The job requires electricians, carpenters, and plumbers, and can cost as much as $600 a square foot, compared to $60 a square foot for a bedroom. Summaries for the average cost of major kitchen remodeling jobs in each of a sample of 11 major U.S. cities is shown in the figure below. You are also told that the true standard deviation of mean kitchen remodeling costs per city across all major cities in the U.S. is $2,500.
(A)The Des Moines Register published an article stating that the average cost of kitchen remodeling in the greater Des Moines Area (which is considered a major U.S. city) is $24,600. Can you compute what percentage of major U.S. cities have a mean kitchen remodeling cost lower than Des Moines? If you can, find this percentage and show your work. If not, explain why not.
No. We cannot draw any conclusions about individual cities, since we have not been provided any information about the distribution of the average cost of kitchen remodeling per U.S. major cities.
(B)You would like to estimate the true mean kitchen remodeling cost per city for all U.S.
major cities by constructing a confidence interval. To do so correctly, do you need to make more assumptions than what is provided in the statement of the problem? Choose one: YES or NO.
(C)If your answer in part (A) was “Yes”, list here ALL the additional assumptions needed. Otherwise, explain why you chose “No” for an answer.
To ensure lack of bias, we need a representative sample, so we assume the sample of 11 cities was drawn at random.
Since the sample size was very small, we need to assume the distribution of average costs per U.S. cities is normal.
FOR QUESTIONS D, E and F, ASSUME THAT ALL CONDITIONS REQUIRED TO CONSTRUCT AN ADEQUATE CONFIDENCE INTERVAL ARE SATISFIED.
(D)Calculate a 95% confidence interval for the true mean kitchen remodeling cost per city for all U.S. major cities. Show all your work to receive full credit. ROUND YOUR ANSWERS TO THE NEAREST 2 DECIMAL PLACES.
(E)JMP also provides a confidence interval for the population mean. This interval is expected to be: NARROWER THAN/ WIDER THAN/ AS WIDE AS the interval computed in part (D). Circle the correct answer. (Note that you do not actually need to compute the interval in part (D) to answer this question correctly).
(F)Explain your choice in part (E) without using the results in part (D).
JMP uses the (estimated) sample standard deviation s to compute the interval. When the population standard deviation is known, there is less variability to account for in the confidence interval, and therefore, the interval is more precise (or narrower).
(G)The Histogram, Box Plot and Quantiles from JMP are examples of: (Circle one.)
Descriptive statisticsInferential statistics
(H)The confidence interval in (D) is an example of: (Circle one.)
Descriptive statisticsInferential statistics
6.Circle ALL correct answers below:
(A)Increasing the sample size can make even a small effect practically significant.
(B)At large sample sizes, all statistically significant effects are practically significant.
(C)Statistical significance does not imply that you have made an important or meaningful discovery.
(D)Increasing the sample size can make even a small effect statistically significant.
(E)None of the above.
7.[BONUS] Which type of error? A chemical firm has been accused of polluting the local river system. State laws require the accuser to prove the polluting by a statistical analysis of water samples. Which type of error is the chemical firm worried about?
CIRCLE ONE and explain how you reached your answer:Type IorType II
The null hypothesis here corresponds to the assumption of innocence, i.e. the firm is NOT polluting. A Type I error would conclude that they are polluting (reject H0) when they are, in reality, not polluting (i.e. H0 is true).