1. What is the appropriate critical value for each of the following confidence levels and sample sizes? *(Remember that in regard to confidence level this implies two-tails)
(a) Critical z-value for an 90% confidence level for sample of size n= 65:
(b) Critical t-value for a 98% confidence level for sample of size n = 22:
2. Apple iPod engineers would like to know the mean time for music downloads. They calculated the download time for specific song by 20 randomly selected iPods, and the data is shown below.
Download time of Apple iPods
24 28 27 24 28
23 26 26 25 26
23 25 22 28 25
24 25 24 25 24
Determine the mean and standard deviation of this sample data.
(b) Determine the expected margin of error amount in using this sample result to
estimate the mean download time for this same song of all Apple iPods based upon a 95% desired level of confidence.
3. The posted speed limit on the 405 Hwy. is 75 mph, but its claimed the mean of the actual speeds of cars is greater than 75 mph. To test this claim, the speeds of 36 randomly selected cars were taken and then reported above.
Give the null and alternative hypotheses for this situation in mathematical notation.
Based upon a 5% level of significance, determine the critical value(s) associated with this hypothesis test.
Determine the sample's test statistic.
(d) State a final conclusion regarding the results of the hypothesis test. Make sure your statement is tied to the context of the problem (speed of cars on the 405 Hwy).
Speeds of cars on the 405 Hwy.
73 77 74 82 77 82
68 78 68 72 77 76
74 79 74 79 76 75
81 76 75 81 69 76
75 66 79 72 69 78
84 79 83 75 71 76
4. A statistical hypothesis test is used to test a claim. You get 1.96 as the test statistic from the collected sample and a 1.85 as your critical value in a right-tailed test. Which of the following is the correct decision statement for the test?
A. Fail to reject the null hypothesis
B. Claim the null hypothesis is true
C. Reject the null hypothesis
D. Claim the alternative hypothesis is false
5. A statistical hypothesis test is used to test a claim. You get a P-value of 0.083 on a test with a significance level of 0.05. Which of the following is the correct decision statement for the test?
A. Fail to reject the null hypothesis
B. Claim the null hypothesis is true
C. Reject the null hypothesis
D. Claim the alternative hypothesis is false
6. A report states that 85% of MLB fans want a larger All Star Game FanFest. One expert says college students don't agree, so she randomly selects 125 NYU college students and finds that 93 want a larger tournament. Follow the steps below to conduct a hypothesis test to determine, at the 10% significance level, if the percentage of students wanting a larger tournament is less than 85%.
(a) List the null and alternative hypotheses for this test.
Determine the value of the sample's test statistic.
Determine the critical value(s).
Write a final interpretive sentence tied to the context for the hypothesis test results.
7. Self-reported heights and measured heights of twelve males aged 12-16 are shown in the table to the right. At the 5% significance level, is there sufficient evidence to support the claim that there is a difference between the average reported heights and the measured heights of males aged 12-16? All measurements are in inches. (Note: Hypothesis testing method work must be shown for credit on this problem);
Reported Height Measured Height
69 67.9
71 69.9
64 64.9
71 68.3
72 70.3
61 60.6
65 64.5
67 67.0
58 55.6
65 64.2
66 65.0
70 70.8
8. Assume that the prices of regular unleaded gas across the nation are normally distributed with a mean of $2.95 and a standard deviation of $0.28.
(a) Describe the shape and horizontal scaling on the graph of the distribution for the population of all regular unleaded gas prices.
(b) If all possible samples of size 25 from the population of these gas prices are drawn and the mean is found for each sample, describe the shape and horizontal scaling on the graph of the sampling distribution for these sample mean values as theorized by the Central Limit Theorem.
(c) Find the probability that the price from a single randomly selected gas station will be more than $3.30. Based upon your results, would it be unusual to find 1 gas station where the price is more than $3.30? Explain.
9. A 95 % confidence interval for a situation is developed and is given as follows: 55
(a) Give the value of the sample mean that was used in developing this interval.
(b) Give the value of the margin of error that was used in developing this interval.
10. A survey of 200 registered voters revealed that 119 of them plan on voting for the incumbent. Find a 99% confidence interval for the proportion of all registered voters who plan on voting for the incumbent. Does it appear statistically valid to conclude that more than 50% of all registered voters plan on voting for the incumbent? Why or why not?
11. Suppose you want to estimate the percentage of registered voters who plan to vote for the incumbent and you want your estimate to be within 4 percentage points of the correct population measure, based upon a 95% confidence level. What minimum size sample is required? Assume that no estimate of "p-hat" is known.
12. A sample of 120 U.S. households has a mean computer usage time of 12.2 hours per week with a sample standard deviation of 2.3 hours. Find a 95% confidence interval for the mean computer usage time of all U.S. households.
13. A social worker is concerned about the number of prescriptions elderly patients take per day. She would like to create a 90% confidence interval with a maximum error of 0.25 prescriptions/day. Assuming 3.2 is a good estimate of the population standard deviation, what is the minimum number of patients she must sample?
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***Multiple Choice:
14. Out of fifty randomly chosen adults age 65 or older, at least 15 had a Facebook account. What is the complement of this description?
A. Fifteen or fewer had a Facebook account.
B. Fewer than fifteen had a Facebook account.
C. No more than fifteen had a Facebook account.
D. Fifteen or more had a Facebook account.
15. If P(A) = 0.08, which one of the following statements is true?
A. The probability of the complement of event A is 0.32.
B. The probability of event A happening twice in a row (with replacement) is 0.0064.
C. Event A is an “unusual” event
D. For each 8 times event A happens, there are 100 times in which A doesn’t happen.
16. A bag contains 6 red marbles, 4 blue marbles, and 3 green marbles. If you draw two marbles, without replacement, what is the probability that you get 2 red marbles?
A. 0.1775 B. 0.1923 C. 0.8782 D. 0.8462
17. A widely accepted fact is that 80% of all people have brown eyes. What is the probability of randomly selecting 2 people and neither of them has brown eyes?
A. 1.6 B. 0.04 C. 0.64 D. 0.16
Problems 18 through 24 refer to the following:
Daily temperatures in Honolulu are normally distributed with a mean of 73 degrees and a standard deviation of 5 degrees.
18. What is the z-score that corresponds to a temperature of 80 degrees within this distribution?
A. −1.4 B. 1.4 C. 0.85 D. 1.03
19. What symmetric interval about the mean will contain approximately 99.7% of the daily temperatures?
A. 58 to 88 B. 63 to 83 C. 68 to 78 D. 53 to 93
20. What is the probability that a randomly selected day will have a temperature below 65 degrees?
A. 0.95 B. −1.6 C. 1.6 D. 0.05
21. What temperature is at the 40th percentile (rounded to the nearest whole number)?
A. 72 B. 74 C. 69 D. 29
22. What percentage of daily temperatures are between 75 degrees and 80 degrees?
A. 0.34 B. 0.26 C. 0.84 D. 0.08
23. Suppose random samples of 25 daily temperatures are selected repeatedly from the population. What is the mean and standard deviation for the sampling distribution of sample means?
A. Mean = 73, SD = 10
B. Mean = 73, SD = 25
C. Mean = 73, SD = 5
D. Mean = 73, SD = 1
24. What is the probability that a sample of 25 daily temperatures chosen at random have an average above 75 degrees?
A. 0.9773 B. 0.3446 C. 0.0228 D. 0.6554
Problems 25 through 27 are based on the information provided directly below:
Suppose you want to determine the average GPA of all OSU students. You randomly select 33 students and find that those students had a mean GPA of 2.68.
25. Compute a 99% confidence interval for the mean GPA of all OSU students, if the sample standard deviation was s = 0.9.
A. 2.68 ± 0.372 B. 2.68 ± 0.429 C. 2.68 ± 0.404 D. 2.68 ± 0.451
26. Assume the population standard deviation is estimated to be σ = 0.8. What sample size, n, is needed to obtain a margin of error of 0.25 with 99% confidence? A. 68 B. 49 C. 30 D. 56
27. Consider the situation given in Problem #26. Which of the following would produce a confidence interval with a larger margin of error?
A. Using a confidence level of 90%
B. Using a smaller estimate for σ
C. Using a smaller sample size
D. All of the above A through C
28. Which of the following statements is false?
A. The Central Limit Theorem states that a sampling distribution of means will not have the same shape as the population distribution from which it is taken.
B. The Central Limit Theorem states that the mean of a sampling distribution of meanswill have the same mean as the population distribution from which it is taken.
C. The Central Limit Theorem states that the standard deviation of a sampling distribution of means (with n > 1) will have a smaller standard deviation as the population distribution from which it is taken.
D. None of A through C are false.
29. If your population is normally distributed, which of the following statements regarding confidence intervals of population means is always true?
A. When σ is known, we use the critical value, z.
B. When n >30, we use the critical value, z.
C. When n < 30, we use the critical value, t.D. None of A through C are always true.
30. A researcher is interested in conducting a poll on the President’s approval rating. A margin of error of at most 2% is desired. How many people must be sampled to meet this requirement at the 95% confidence level if no preliminary estimate of p-hat is known?
A. 1691
B. 250
C. 2401
D. 170
31. The Pew Research Center conducted a survey of 1007 randomly selected adults and found that 791 of those adults know what Twitter is. Use a 0.01 significance level to test the claim that more than 75% of adults know what Twitter is. Compute the P value for this test.
A. 0.0001
B. 2.76
C. 2.6
D. 0.0046
32. Suppose for a particular hypothesis test, α = 0.05 and the P value = 0.10. Which of the following statements is false?
A. We reject the null hypothesis.
B. We fail to reject the null hypothesis.
C. The observed result is "not unusual".
D. The computed test statistic, z, does not fall in the shaded critical region of the tail in the normal curve.
33. A sample of 40 women is obtained, and their heights (in inches) and pulse rates (in beats per minute) are measured. The average height was 67.9 inches and the average pulse rate was 85.2 bpm. The linear correlation coefficient is r = 0.802 and the equation of the regression line is found to be ŷ = 18.26 + 0.920x , where x represents height. Find the best predicted pulse rate of a woman who is 70 inches tall.
A. 85.2 bpm
B. 82.6 bpm
C. 54.0 bpm
D. 67.9 bpm
34. A correlation coefficient of −0.96 between two quantitative variables A and B indicates that
A. As A increases, B tends to increase.
B. Changes in A cause changes in B.
C. As A increases, B tends to decrease.
D. There is a very weak association between A and B, and change in A will not affect B.
Problems 35 and 36 refer to the following table pairing the values of the Consumer Price Index (CPI) and the national average cost of a slice of pizza:
CPI 30.2 48.3 112.3 162.2 191.9 197.8
Pizza Cost 0.15 0.35 1 1.25 1.75 2
35. Compute the least squares regression line for the cost of pizza.
A. ŷ = − 0.1616x − 0.0101
B. ŷ = 0.0101x − 0.1616
C. ŷ = − 0.0101x + 0.1616
D. ŷ = 0.1616x − 0.0101
36. Calculate the linear correlation coefficient between the two variables.
A. 0.985
B. −0.985
C. 0.971
D. −0.971
