Q1: A researcher wants to compare the proportion of students who prefer online learning in two different schools. School A has 200 students with 80 preferring online learning, and School B has 150 students with 45 preferring online learning. What is the sample proportion for School A? (a) 0.30 (b) 0.40 (c) 0.50 (d) 0.625
Solution:
Ans: (b) Explanation: The sample proportion is calculated as \(\hat{p}_A = \frac{x}{n}\), where \(x\) is the number of successes and \(n\) is the sample size. For School A: \(\hat{p}_A = \frac{80}{200} = 0.40\). Option (a) is the proportion for School B, option (c) is a common miscalculation, and option (d) results from inverting the calculation.
Q2: When comparing two proportions, which of the following is the correct formula for the pooled sample proportion? (a) \(\hat{p}_{pool} = \frac{x_1 + x_2}{n_1 + n_2}\) (b) \(\hat{p}_{pool} = \frac{x_1 - x_2}{n_1 - n_2}\) (c) \(\hat{p}_{pool} = \frac{\hat{p}_1 + \hat{p}_2}{2}\) (d) \(\hat{p}_{pool} = \frac{n_1 + n_2}{x_1 + x_2}\)
Solution:
Ans: (a) Explanation: The pooled sample proportion combines the successes from both samples divided by the total sample size: \(\hat{p}_{pool} = \frac{x_1 + x_2}{n_1 + n_2}\). Option (b) uses subtraction incorrectly, option (c) is the average of proportions (not the pooled proportion), and option (d) inverts the correct formula.
Q3: A hypothesis test is conducted to compare two proportions. The null hypothesis states: (a) \(p_1 > p_2\) (b) \(p_1 = p_2\) (c) \(p_1 <> (d) \(p_1 \neq p_2\)
Solution:
Ans: (b) Explanation: In hypothesis testing for comparing two proportions, the null hypothesis typically states that there is no difference between the populations: \(H_0: p_1 = p_2\). Options (a) and (c) represent one-sided alternative hypotheses, while option (d) represents a two-sided alternative hypothesis.
Q4: Two groups are being compared. Group 1 has \(\hat{p}_1 = 0.55\) with \(n_1 = 100\), and Group 2 has \(\hat{p}_2 = 0.45\) with \(n_2 = 100\). What is the difference in sample proportions \(\hat{p}_1 - \hat{p}_2\)? (a) 0.10 (b) 1.00 (c) 0.50 (d) -0.10
Solution:
Ans: (a) Explanation: The difference in sample proportions is calculated as \(\hat{p}_1 - \hat{p}_2 = 0.55 - 0.45 = 0.10\). Option (b) results from multiplication instead of subtraction, option (c) is the average of the proportions, and option (d) reverses the order of subtraction.
Q5: When constructing a confidence interval for the difference between two proportions, which condition must be satisfied? (a) Both sample sizes must be exactly equal (b) \(n_1\hat{p}_1 \geq 10\), \(n_1(1-\hat{p}_1) \geq 10\), \(n_2\hat{p}_2 \geq 10\), and \(n_2(1-\hat{p}_2) \geq 10\) (c) The populations must be infinite (d) The difference must be positive
Solution:
Ans: (b) Explanation: The success-failure condition requires that \(n_1\hat{p}_1 \geq 10\), \(n_1(1-\hat{p}_1) \geq 10\), \(n_2\hat{p}_2 \geq 10\), and \(n_2(1-\hat{p}_2) \geq 10\) to ensure the sampling distribution is approximately normal. Option (a) is unnecessary, option (c) is not a requirement (though samples should be less than 10% of population), and option (d) is incorrect as differences can be negative.
Q6: A two-proportion z-test yields a test statistic of \(z = 2.5\). Using a significance level of 0.05 for a two-tailed test, what conclusion should be drawn? (a) Fail to reject the null hypothesis (b) Reject the null hypothesis (c) Accept the alternative hypothesis as fact (d) The test is inconclusive
Solution:
Ans: (b) Explanation: For a two-tailed test at \(\alpha = 0.05\), the critical values are approximately \(\pm 1.96\). Since \(z = 2.5\) falls in the rejection region (beyond 1.96), we reject the null hypothesis. Option (a) would apply if \(|z| < 1.96\),="" option="" (c)="" uses="" incorrect="" language="" (we="" don't="" "accept"="" hypotheses),="" and="" option="" (d)="" is="" incorrect="" as="" the="" test="" provides="" clear="" evidence.="">
Q7: The standard error for the difference between two sample proportions \(\hat{p}_1 - \hat{p}_2\) is given by: (a) \(\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}\) (b) \(\frac{\hat{p}_1 - \hat{p}_2}{n_1 + n_2}\) (c) \(\sqrt{\frac{\hat{p}_1 + \hat{p}_2}{n_1 + n_2}}\) (d) \(\hat{p}_1(1-\hat{p}_1) + \hat{p}_2(1-\hat{p}_2)\)
Solution:
Ans: (a) Explanation: The standard error for the difference between two independent proportions is \(SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}\). This formula accounts for variability in both samples. Options (b), (c), and (d) do not represent valid standard error formulas for comparing proportions.
Q8: A 95% confidence interval for \(p_1 - p_2\) is calculated as (-0.05, 0.15). What can be concluded? (a) There is significant evidence that \(p_1 > p_2\) (b) There is significant evidence that \(p_1 <> (c) There is not sufficient evidence to conclude that \(p_1 \neq p_2\) (d) The proportions are definitely equal
Solution:
Ans: (c) Explanation: Since the confidence interval contains 0 (ranging from -0.05 to 0.15), we cannot conclude that there is a significant difference between \(p_1\) and \(p_2\) at the 0.05 significance level. Options (a) and (b) would require the interval to be entirely positive or negative, respectively. Option (d) is incorrect because we never prove equality, only fail to find evidence of difference.
Section B: Fill in the Blanks
Q9: When testing \(H_0: p_1 = p_2\), the test statistic follows a __________ distribution when the null hypothesis is true.
Solution:
Ans: standard normal (or z) Explanation: Under the null hypothesis and when conditions are met, the test statistic for comparing two proportions follows a standard normal distribution (z-distribution).
Q10: In a two-proportion z-test, if the p-value is less than the significance level \(\alpha\), we __________ the null hypothesis.
Solution:
Ans: reject Explanation: When the p-value is less than the significance level \(\alpha\), there is sufficient evidence to reject the null hypothesis.
Q11: The formula for the test statistic when comparing two proportions is \(z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{__________}\), where the denominator represents the standard error using the pooled proportion.
Solution:
Ans: \(\sqrt{\hat{p}_{pool}(1-\hat{p}_{pool})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}\) Explanation: The standard error for the test statistic uses the pooled proportion and is calculated as \(\sqrt{\hat{p}_{pool}(1-\hat{p}_{pool})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}\).
Q12: A __________ is a range of plausible values for the difference between two population proportions.
Solution:
Ans: confidence interval Explanation: A confidence interval provides a range of values that likely contains the true difference between two population proportions with a specified level of confidence.
Q13: When samples are selected independently from two populations, the sampling method is called __________ samples.
Solution:
Ans: independent Explanation:Independent samples means that the selection of individuals in one sample does not influence the selection in the other sample, which is a key assumption when comparing two proportions.
Q14: The critical value for a two-tailed test at the 0.05 significance level is approximately __________.
Solution:
Ans: ±1.96 Explanation: For a two-tailed test with \(\alpha = 0.05\), the critical values are approximately \(\pm 1.96\), meaning we reject the null hypothesis if \(z < -1.96\)="" or="" \(z=""> 1.96\).
Section C: Word Problems
Q15: A company wants to compare customer satisfaction between two stores. In Store A, 120 out of 200 customers reported being satisfied. In Store B, 90 out of 180 customers reported being satisfied. Calculate the pooled sample proportion.
Solution:
Ans: Step 1: Identify the values: \(x_1 = 120\), \(n_1 = 200\), \(x_2 = 90\), \(n_2 = 180\) Step 2: Use the pooled proportion formula: \(\hat{p}_{pool} = \frac{x_1 + x_2}{n_1 + n_2}\) Step 3: Calculate: \(\hat{p}_{pool} = \frac{120 + 90}{200 + 180} = \frac{210}{380} = 0.5526\) Final Answer: 0.5526 (or approximately 0.553)
Q16: A researcher is testing whether the proportion of males who support a new policy differs from the proportion of females who support it. In a sample of 150 males, 75 support the policy. In a sample of 200 females, 120 support the policy. Calculate the difference in sample proportions \(\hat{p}_m - \hat{p}_f\).
Solution:
Ans: Step 1: Calculate the sample proportion for males: \(\hat{p}_m = \frac{75}{150} = 0.50\) Step 2: Calculate the sample proportion for females: \(\hat{p}_f = \frac{120}{200} = 0.60\) Step 3: Find the difference: \(\hat{p}_m - \hat{p}_f = 0.50 - 0.60 = -0.10\) Final Answer: -0.10
Q17: Two schools are comparing graduation rates. School X has 180 graduates out of 200 students, and School Y has 162 graduates out of 180 students. Construct a 95% confidence interval for the difference in proportions \(p_X - p_Y\). (Use \(z^* = 1.96\))
Q18: A pharmaceutical company tests a new drug on two groups. In Group 1 (treatment), 85 out of 100 patients showed improvement. In Group 2 (placebo), 60 out of 100 patients showed improvement. Test at the 0.05 significance level whether the treatment proportion is greater than the placebo proportion. Calculate the test statistic.
Q19: In a survey, 45 out of 90 teenagers prefer social media platform A, while 70 out of 120 teenagers prefer social media platform B. Verify whether the success-failure condition is met for constructing a confidence interval.
Solution:
Ans: Step 1: Calculate sample proportions: \(\hat{p}_1 = \frac{45}{90} = 0.50\), \(\hat{p}_2 = \frac{70}{120} \approx 0.583\) Step 2: Check for Group 1: \(n_1\hat{p}_1 = 90(0.50) = 45 \geq 10\) ✓ and \(n_1(1-\hat{p}_1) = 90(0.50) = 45 \geq 10\) ✓ Step 3: Check for Group 2: \(n_2\hat{p}_2 = 120(0.583) \approx 70 \geq 10\) ✓ and \(n_2(1-\hat{p}_2) = 120(0.417) \approx 50 \geq 10\) ✓ Final Answer: Yes, the success-failure condition is met for both groups.
Q20: A political poll shows that 240 out of 400 voters in City A support a candidate, while 180 out of 300 voters in City B support the same candidate. Calculate the standard error for the difference in sample proportions using the individual sample proportions (not pooled).
Previous Year Questions with Solutions, Worksheet (with Solutions): Comparing Two Proportions, Objective type Questions, past year papers, MCQs, practice quizzes, Sample Paper, Exam, shortcuts and tricks, Extra Questions, Important questions, Free, pdf , Semester Notes, Worksheet (with Solutions): Comparing Two Proportions, ppt, study material, Viva Questions, Worksheet (with Solutions): Comparing Two Proportions, mock tests for examination, video lectures, Summary;
