OPRE 6301 Test 3 (Statistics and Data Analysis) Questions
and Answers
Question 1: Hypothesis Testing
A sample of 40 students has a mean exam score of 78 with a standard deviation of 10. Test at the
5% level whether the population mean is different from 75.
• Step 1: Null hypothesis (H_0: \mu = 75), Alternative (H_a: \mu \neq 75).
• Step 2: Test statistic:
[ t = \frac{78 - 75}{10/\sqrt{40}} = \frac{3}{1.58} \approx 1.90 ]
• Step 3: Critical value at df = 39, two-tailed, (\alpha = 0.05) → (t_{crit} \approx \pm
2.02).
• Answer: Fail to reject (H_0) (since 1.90 < 2.02).
Question 2: Confidence Interval
A sample of 64 observations has a mean of 120 and variance of 144. Construct a 95% confidence
interval for the population mean.
• Standard error = (\sqrt{144}/\sqrt{64} = 12/8 = 1.5).
• CI = (120 \pm 1.96(1.5)).
• CI = (120 \pm 2.94).
• Interval = (117.06, 122.94).
Question 3: Regression Analysis
In a regression model predicting sales (Y) from advertising spend (X), the estimated equation is:
[ Y = 50 + 6X ]
If advertising spend = 20, what is the predicted sales?
• Predicted (Y = 50 + 6(20) = 50 + 120 = 170).
• Answer: 170 units.
Question 4: ANOVA
Three teaching methods were tested with sample means: Method A = 82, Method B = 78,
Method C = 85. Assume equal sample sizes and variance. The ANOVA F-statistic is significant
at (\alpha = 0.05). What conclusion can be drawn?
• Answer: At least one teaching method mean differs significantly from the others.
,Question 5: Probability
A company’s defect rate is 4%. If 10 items are inspected, what is the probability that exactly 2
are defective?
• Binomial: (P(X=2) = \binom{10}{2}(0.04)^2(0.96)^8).
• Calculation: (45 \cdot 0.0016 \cdot 0.721).
• ≈ 0.052.
• Answer: Probability ≈ 5.2%.
Question 6: Sampling Distribution
A population has mean 50 and standard deviation 12. A sample of 36 is taken. What is the
standard error of the mean?
• SE = (12/\sqrt{36} = 12/6 = 2).
• Answer: 2.
Question 7: Chi-Square Test
A survey of 100 students shows 40 prefer online classes, 60 prefer in-person. Expected split is
50–50. Test at (\alpha = 0.05).
• (\chi^2 = \frac{(40-50)^2}{50} + \frac{(60-50)^2}{50} = \frac{100}{50} +
\frac{100}{50} = 4).
• df = 1, critical ≈ 3.84.
• Answer: Reject (H_0). Preferences differ significantly.
Question 8: Regression Coefficient
Regression output: slope = 0.8, SE = 0.2, t = ?
• (t = 0.8/0.2 = 4.0).
• Answer: t = 4.0.
Question 9: Probability
A die is rolled twice. Probability of sum = 7?
• Outcomes = 36. Favorable = 6 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1).
• (6/36 = 1/6).
, • Answer: 0.167 (16.7%).
Question 10: ANOVA
Four groups tested, F-statistic = 5.2, critical F = 3.1.
• Answer: Reject (H_0). At least one group mean differs.
Question 11: Confidence Interval
Sample mean = 200, n = 25, s = 20. 95% CI?
• SE = 20/√25 = 4.
• CI = 200 ± 1.96(4) = 200 ± 7.84.
• Interval = (192.16, 207.84).
• Answer: (192.16, 207.84).
Question 12: Hypothesis Test
Population mean = 100, sample mean = 104, n = 64, s = 16. Test at 1%.
• SE = 16/√64 = 2.
• t = (104–100)/2 = 2.0.
• Critical t ≈ 2.64.
• Answer: Fail to reject (H_0).
Question 13: Probability
If P(A) = 0.4, P(B) = 0.5, independent, find P(A ∩ B).
• 0.4 × 0.5 = 0.2.
• Answer: 0.2.
Question 14: Regression Prediction
Equation: Y = 10 + 2X. If X = 15, Y = ?
• Y = 10 + 30 = 40.
• Answer: 40.
and Answers
Question 1: Hypothesis Testing
A sample of 40 students has a mean exam score of 78 with a standard deviation of 10. Test at the
5% level whether the population mean is different from 75.
• Step 1: Null hypothesis (H_0: \mu = 75), Alternative (H_a: \mu \neq 75).
• Step 2: Test statistic:
[ t = \frac{78 - 75}{10/\sqrt{40}} = \frac{3}{1.58} \approx 1.90 ]
• Step 3: Critical value at df = 39, two-tailed, (\alpha = 0.05) → (t_{crit} \approx \pm
2.02).
• Answer: Fail to reject (H_0) (since 1.90 < 2.02).
Question 2: Confidence Interval
A sample of 64 observations has a mean of 120 and variance of 144. Construct a 95% confidence
interval for the population mean.
• Standard error = (\sqrt{144}/\sqrt{64} = 12/8 = 1.5).
• CI = (120 \pm 1.96(1.5)).
• CI = (120 \pm 2.94).
• Interval = (117.06, 122.94).
Question 3: Regression Analysis
In a regression model predicting sales (Y) from advertising spend (X), the estimated equation is:
[ Y = 50 + 6X ]
If advertising spend = 20, what is the predicted sales?
• Predicted (Y = 50 + 6(20) = 50 + 120 = 170).
• Answer: 170 units.
Question 4: ANOVA
Three teaching methods were tested with sample means: Method A = 82, Method B = 78,
Method C = 85. Assume equal sample sizes and variance. The ANOVA F-statistic is significant
at (\alpha = 0.05). What conclusion can be drawn?
• Answer: At least one teaching method mean differs significantly from the others.
,Question 5: Probability
A company’s defect rate is 4%. If 10 items are inspected, what is the probability that exactly 2
are defective?
• Binomial: (P(X=2) = \binom{10}{2}(0.04)^2(0.96)^8).
• Calculation: (45 \cdot 0.0016 \cdot 0.721).
• ≈ 0.052.
• Answer: Probability ≈ 5.2%.
Question 6: Sampling Distribution
A population has mean 50 and standard deviation 12. A sample of 36 is taken. What is the
standard error of the mean?
• SE = (12/\sqrt{36} = 12/6 = 2).
• Answer: 2.
Question 7: Chi-Square Test
A survey of 100 students shows 40 prefer online classes, 60 prefer in-person. Expected split is
50–50. Test at (\alpha = 0.05).
• (\chi^2 = \frac{(40-50)^2}{50} + \frac{(60-50)^2}{50} = \frac{100}{50} +
\frac{100}{50} = 4).
• df = 1, critical ≈ 3.84.
• Answer: Reject (H_0). Preferences differ significantly.
Question 8: Regression Coefficient
Regression output: slope = 0.8, SE = 0.2, t = ?
• (t = 0.8/0.2 = 4.0).
• Answer: t = 4.0.
Question 9: Probability
A die is rolled twice. Probability of sum = 7?
• Outcomes = 36. Favorable = 6 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1).
• (6/36 = 1/6).
, • Answer: 0.167 (16.7%).
Question 10: ANOVA
Four groups tested, F-statistic = 5.2, critical F = 3.1.
• Answer: Reject (H_0). At least one group mean differs.
Question 11: Confidence Interval
Sample mean = 200, n = 25, s = 20. 95% CI?
• SE = 20/√25 = 4.
• CI = 200 ± 1.96(4) = 200 ± 7.84.
• Interval = (192.16, 207.84).
• Answer: (192.16, 207.84).
Question 12: Hypothesis Test
Population mean = 100, sample mean = 104, n = 64, s = 16. Test at 1%.
• SE = 16/√64 = 2.
• t = (104–100)/2 = 2.0.
• Critical t ≈ 2.64.
• Answer: Fail to reject (H_0).
Question 13: Probability
If P(A) = 0.4, P(B) = 0.5, independent, find P(A ∩ B).
• 0.4 × 0.5 = 0.2.
• Answer: 0.2.
Question 14: Regression Prediction
Equation: Y = 10 + 2X. If X = 15, Y = ?
• Y = 10 + 30 = 40.
• Answer: 40.
