Practice Final Exam Questions (2) -- Answers
Part A. Multiple Choice Questions. For each question, you are encouraged to give a reason or show work for
partial credit. You must show your work or reason if the question is marked with an asterisk (*).
1. Confidence intervals are useful when trying to estimate _______.
a. unknown parameters
b. known parameters
c. unknown statistics
d. known statistics
2. The one-sample z statistic is used instead of the one-sample t statistic when ______.
a. μ is known
b. μ is unknown
c. σ is known
d. σ is unknown
3. The ____ the P-value, the stronger the evidence against the null hypothesis provided by the data.
a. larger
b. smaller
4. (*) The test statistic for a two-sided significance test for a population mean is z = –2.12. What is the
corresponding P-value?
a. 0.017
b. 0.034
c. 0.483
d. 0.983
Answer: P(Z < -2.12) = 0.0170. Since this is a two-sided test, P-value is twice as much to include the other
tail. 0.0170 * 2 = 0.034.
5. The probability you reject the null hypothesis when in fact the null hypothesis is true is called __________.
a. a Type I error
b. a Type II error
c. the power
6. (*) A random sample of 20 observations produced a sample mean of 𝑥𝑥̅ = 92.4 and s = 25.8. What is the
value of the standard error of 𝑥𝑥̅ ?
a. 4.6
b. 15.9
c. 1.3
d. 5.8
e. 2.6
25.8 25.8
Answer: 𝑆𝑆𝑆𝑆𝑥𝑥̅ = = 4.472 = 5.8
√20
, 7. (*) The heights (in inches) of adult males in the United States are believed to be Normally distributed with
mean µ. The average height of a random sample of 25 American adult males is found to be 𝑥𝑥 ̅ = 69.72
inches, and the standard deviation of the 25 heights is found to be s = 4.15. A 90% confidence interval for µ is
a. 69.72 ± 1.09
b. 69.72 ± 1.37
c. 69.72 ± 1.42
Answer: The t critical value for 90% CI for df = 24 is 1.711. So
4.15
margin_or_error = (1.711) ∙ = (1.711) ∙ (0.83) = 1.42
√25
8. Suppose we were interested in determining if there were differences in the average prices among two
local supermarkets. We randomly pick six items to compare at both supermarkets. Which statistical
procedure would be best to use for this study?
a. Matched-pairs t procedure if you interpreted as ‘same six items’
b. One-sample t test
c. Two-sample t test if you interpreted as ‘different six items’
d. None of the above
9. (*) Perform a one-sample t-test using the following statistics:
n=5 𝑥𝑥̅ = 3.871 s = 0.679
The null hypothesis is μ = 5.0 is
a. accepted at the 5% level; accepted at the 1% level.
b. accepted at the 5% level; rejected at the 1% level.
c. rejected at the 5% level; accepted at the 1% level.
d. rejected at the 5% level; rejected at the 1% level.
𝑥𝑥̅ −𝜇𝜇 3.871−5.0
Answer: Test statistic: 𝑡𝑡 = 𝑠𝑠 = 0.679 = −3.71
√𝑛𝑛 √5
Assuming a two-sided test, for df = 4,
• 95% CI of the t-statistic assuming H0 is [-2.776, 2.776]
• 99% CI of the t-statistic assuming H0 is [-4.604, 4.604]
Therefore we reject H0 for alpha = 0.05 but not for alpha 0.01.
10. (*) You buy a package of 122 Smarties and 19 of them are red. What is a 95% confidence interval for the
true proportion of red Smarties?
a. (0.092, 0.220)
b. (0.103, 0.230)
c. (0.085, 0.199)
19 (0.156)∙(0.844)
Answer: 𝑝𝑝̂ = 122 = 0.156. margin_of_error = (1.96) ∙ � = (1.96) ∙ (0.033) = 0.0647.
122
So the 95% CI is [0.156 – 0.0647, 0.156 + 0.0647], which yields [0.0913, 0.2207].
Part A. Multiple Choice Questions. For each question, you are encouraged to give a reason or show work for
partial credit. You must show your work or reason if the question is marked with an asterisk (*).
1. Confidence intervals are useful when trying to estimate _______.
a. unknown parameters
b. known parameters
c. unknown statistics
d. known statistics
2. The one-sample z statistic is used instead of the one-sample t statistic when ______.
a. μ is known
b. μ is unknown
c. σ is known
d. σ is unknown
3. The ____ the P-value, the stronger the evidence against the null hypothesis provided by the data.
a. larger
b. smaller
4. (*) The test statistic for a two-sided significance test for a population mean is z = –2.12. What is the
corresponding P-value?
a. 0.017
b. 0.034
c. 0.483
d. 0.983
Answer: P(Z < -2.12) = 0.0170. Since this is a two-sided test, P-value is twice as much to include the other
tail. 0.0170 * 2 = 0.034.
5. The probability you reject the null hypothesis when in fact the null hypothesis is true is called __________.
a. a Type I error
b. a Type II error
c. the power
6. (*) A random sample of 20 observations produced a sample mean of 𝑥𝑥̅ = 92.4 and s = 25.8. What is the
value of the standard error of 𝑥𝑥̅ ?
a. 4.6
b. 15.9
c. 1.3
d. 5.8
e. 2.6
25.8 25.8
Answer: 𝑆𝑆𝑆𝑆𝑥𝑥̅ = = 4.472 = 5.8
√20
, 7. (*) The heights (in inches) of adult males in the United States are believed to be Normally distributed with
mean µ. The average height of a random sample of 25 American adult males is found to be 𝑥𝑥 ̅ = 69.72
inches, and the standard deviation of the 25 heights is found to be s = 4.15. A 90% confidence interval for µ is
a. 69.72 ± 1.09
b. 69.72 ± 1.37
c. 69.72 ± 1.42
Answer: The t critical value for 90% CI for df = 24 is 1.711. So
4.15
margin_or_error = (1.711) ∙ = (1.711) ∙ (0.83) = 1.42
√25
8. Suppose we were interested in determining if there were differences in the average prices among two
local supermarkets. We randomly pick six items to compare at both supermarkets. Which statistical
procedure would be best to use for this study?
a. Matched-pairs t procedure if you interpreted as ‘same six items’
b. One-sample t test
c. Two-sample t test if you interpreted as ‘different six items’
d. None of the above
9. (*) Perform a one-sample t-test using the following statistics:
n=5 𝑥𝑥̅ = 3.871 s = 0.679
The null hypothesis is μ = 5.0 is
a. accepted at the 5% level; accepted at the 1% level.
b. accepted at the 5% level; rejected at the 1% level.
c. rejected at the 5% level; accepted at the 1% level.
d. rejected at the 5% level; rejected at the 1% level.
𝑥𝑥̅ −𝜇𝜇 3.871−5.0
Answer: Test statistic: 𝑡𝑡 = 𝑠𝑠 = 0.679 = −3.71
√𝑛𝑛 √5
Assuming a two-sided test, for df = 4,
• 95% CI of the t-statistic assuming H0 is [-2.776, 2.776]
• 99% CI of the t-statistic assuming H0 is [-4.604, 4.604]
Therefore we reject H0 for alpha = 0.05 but not for alpha 0.01.
10. (*) You buy a package of 122 Smarties and 19 of them are red. What is a 95% confidence interval for the
true proportion of red Smarties?
a. (0.092, 0.220)
b. (0.103, 0.230)
c. (0.085, 0.199)
19 (0.156)∙(0.844)
Answer: 𝑝𝑝̂ = 122 = 0.156. margin_of_error = (1.96) ∙ � = (1.96) ∙ (0.033) = 0.0647.
122
So the 95% CI is [0.156 – 0.0647, 0.156 + 0.0647], which yields [0.0913, 0.2207].
