9-5. Exercises

1. BASIC

Ex 1. Estimate the common sample size $n$ of equally sized independent samples needed to estimate $μ_1−μ_2$ as specified when the population standard deviations are as shown.

1. 90% confidence, to within 3 units, $σ_1=10$ and $σ_2=7$

2. 99% confidence, to within 4 units, $σ_1=6.8$ and $σ_2=9.3$

3. 95% confidence, to within 5 units, $σ_1=22.6$ and $σ_2=31.8$

Ex 2. Estimate the common sample size $n$ of equally sized independent samples needed to estimate $μ_1−μ_2$ as specified when the population standard deviations are as shown.

1. 80% confidence, to within 2 units, $σ_1=14$ and $σ_2=23$

2. 90% confidence, to within 0.3 units, $σ_1=1.3$ and $σ_2=0.8$

3. 99% confidence, to within 11 units, $σ_1=42$ and $σ_2=37$

Ex 3. Estimate the number $n$ of pairs that must be sampled in order to estimate $μ_d=μ_1−μ_2$ as specified when the standard deviation sd of the population of differences is as shown.

1. 80% confidence, to within 6 units, $σ_d=26.5$

2. 95% confidence, to within 4 units, $σ_d=12$

3. 90% confidence, to within 5.2 units, $σ_d=11.3$

Ex 4. Estimate the number $n$ of pairs that must be sampled in order to estimate $μ_d=μ_1−μ_2$ as specified when the standard deviation sd of the population of differences is as shown.

1. 90% confidence, to within 20 units, $σ_d=75.5$

2. 95% confidence, to within 11 units, $σ_d=31.4$

3. 99% confidence, to within 1.8 units, $σ_d=4$

Ex 5. Estimate the minimum equal sample sizes $n_1=n_2$ necessary in order to estimate $p_1−p_2$ as specified.

1. 80% confidence, to within 0.05 (five percentage points)

   1. when no prior knowledge of $p_1$ or $p_2$ is available

   2. when prior studies indicate that $p_1≈0.20$ and $p2≈0.65$

2. 90% confidence, to within 0.02 (two percentage points)

   1. when no prior knowledge of $p_1$ or $p_2$ is available

   2. when prior studies indicate that $p_1≈0.75$ and $p_2≈0.63$

3. 95% confidence, to within 0.10 (ten percentage points)

   1. when no prior knowledge of $p_1$ or $p_2$ is available

   2. when prior studies indicate that $p_1≈0.11$ and $p_2≈0.37$

Ex 6. Estimate the minimum equal sample sizes $n_1=n_2$ necessary in order to estimate $p_1−p_2$ as specified.

1. 80% confidence, to within 0.02 (two percentage points)

   1. when no prior knowledge of $p_1$ or $p_2$ is available

   2. when prior studies indicate that $p_1≈0.78$ and $p_2≈0.65$

2. 90% confidence, to within 0.05 (two percentage points)

   1. when no prior knowledge of $p_1$ or $p_2$ is available

   2. when prior studies indicate that $p_1≈0.12$ and $p_2≈0.24$

3. 95% confidence, to within 0.10 (ten percentage points)

   1. when no prior knowledge of $p_1$ or $p_2$ is available

   2. when prior studies indicate that $p_1≈0.14$ and $p_2≈0.21$

2. APPLICATIONS

Ex 7. An educational researcher wishes to estimate the difference in average scores of elementary school children on two versions of a 100-point standardized test, at 99% confidence and to within two points. Estimate the minimum equal sample sizes necessary if it is known that the standard deviation of scores on different versions of such tests is 4.9.

Ex 8. A university administrator wishes to estimate the difference in mean grade point averages among all men affiliated with fraternities and all unaffiliated men, with 95% confidence and to within 0.15. It is known from prior studies that the standard deviations of grade point averages in the two groups have common value 0.4. Estimate the minimum equal sample sizes necessary to meet these criteria.

Ex 9. An automotive tire manufacturer wishes to estimate the difference in mean wear of tires manufactured with an experimental material and ordinary production tire, with 90% confidence and to within 0.5 mm. To eliminate extraneous factors arising from different driving conditions the tires will be tested in pairs on the same vehicles. It is known from prior studies that the standard deviations of the differences of wear of tires constructed with the two kinds of materials is 1.75 mm. Estimate the minimum number of pairs in the sample necessary to meet these criteria.

Ex 10. To assess to the relative happiness of men and women in their marriages, a marriage counselor plans to administer a test measuring happiness in marriage to n randomly selected married couples, record the their test scores, find the differences, and then draw inferences on the possible difference. Let $μ_1$ and $μ_2$ be the true average levels of happiness in marriage for men and women respectively as measured by this test. Suppose it is desired to find a 90% confidence interval for estimating $μ_d=μ_1−μ_2$ to within two test points. Suppose further that, from prior studies, it is known that the standard deviation of the differences in test scores is $σ_d≈10$ . What is the minimum number of married couples that must be included in this study?

Ex 11. A journalist plans to interview an equal number of members of two political parties to compare the proportions in each party who favor a proposal to allow citizens with a proper license to carry a concealed handgun in public parks. Let $p_1$ and $p_2$ be the true proportions of members of the two parties who are in favor of the proposal. Suppose it is desired to find a 95% confidence interval for estimating $(p_1−p_2)$ to within 0.05. Estimate the minimum equal number of members of each party that must be sampled to meet these criteria.

Ex 12. A member of the state board of education wants to compare the proportions of National Board Certified (NBC) teachers in private high schools and in public high schools in the state. His study plan calls for an equal number of private school teachers and public school teachers to be included in the study. Let $p_1$ and $p_2$ be these proportions. Suppose it is desired to find a 99% confidence interval that estimates $(p_1−p_2)$ to within 0.05.

1. Supposing that both proportions are known, from a prior study, to be approximately 0.15, compute the minimum common sample size needed.

2. Compute the minimum common sample size needed on the supposition that nothing is known about the values of $p_1$ and $p_2$ .