Statistics Cheatsheet 4

by Lei Bao

Confidence Intervals and Sample Size

1. A survey of 2302 adults in a certain large country aged 18 and older conducted by a reputable polling organization found that 407 have donated blood in the past two years.

• Obtain a point estimate for the population proportion of adults in the country aged 18 and older who have donated blood in the past two years.

  • Sample Proportion

  • $\hat{p} = \frac{x}{n}$

  • $\hat{p} = \frac{407}{2302} = 0.177$

• Verify that the requirements for constructing a confidence interval about p are satisfied.

  • The sample can be assumed to be a simple random sample, the value of $n \hat{p} ( 1 - \hat{p})$ is 335.335, which is greater than or equal to 10, and the sample size can be assumed to be less than or equal to 5% of the population size.

• Construct and interpret a 90% confidence interval for the population proportion of adults in the country who have donated blood in the past two years.

  • A $( 1- \alpha) \cdot 100\%$ confidence interval about $p$ is

  • $\hat{p} \pm z_{\frac{\alpha}{2}} \cdot \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$

  • Upper Bound: $0.177 + 1.645 \times \sqrt{\frac{0.177 (1-0.177)}{2302}} = 0.190$

  • Lower Bound: $0.177 - 1.645 \times \sqrt{\frac{0.177 (1-0.177)}{2302}} = 0.164$

  • We are 90% confident the proportion of adults in the country aged 18 and older who have donated blood in the past two years in between 0.164 and 0.190.

2. A researcher wishes to estimate the percentage of adults who support abolishing the penny. What size sample should be obtained if he wishes the estimate to be within 5 percentage points with 90% confidence if

• he uses a previous estimate of 34%

  • To estimate the population with a margin of error E at a $(1-\alpha) \cdot 100\%$ level confidence

  • $n = \hat{p} (1- \hat{p}) (\frac{z_{\frac{\alpha}{2}}}{E})^2$ rounded up to the next integer, where $\hat{p}$ is a prior estimate of the population proportion.

  • $n = 0.34 (1- 0.34) (\frac{1.645}{0.05})^2 = 243$

• he does not use any prior estimates?

  • $n = 0.25 ( \frac{z_{\frac{\alpha}{2}}}{E})^2$ rounded up to the next integer when no prior estimate of $p$ is available.

  • $n = 0.25 ( \frac{1.645}{0.05})^2 = 271$

3. A survey was conducted that asked 974 people how many books they had read in the past year. Results indicated the $\bar{x} = 14.5$ books and $s = 15.9$ books. Construct a 98% confidence interval for the mean number of books people read. Interpret the interval.

   • A $(1- \alpha) \cdot 100$ confidence interval about $\mu$ is

   • $\bar{x} \pm t_{\frac{\alpha}{2}} \cdot \frac{s}{\sqrt{n}}$

   • T table

   • $t_{\frac{\alpha}{2}} = 2.364$

   • Upper Bound: $14.5 + 2.364 \cdot \frac{15.9}{\sqrt{974}} = 15.70$

   • Lower Bound: $14.5 - 2.364 \cdot \frac{15.9}{\sqrt{974}} = 13.30$

4. A doctor wants to estimate the mean HDL cholesterol of all 20- to 29-year-old females. How many subjects are needed to estimate the mean HDL cholesterol within 2 points with 99% confidence assuming s = 15.9 based on earlier studies? Suppose the doctor would be content with 95% confidence. How does the decrease in confidence affect the sample size required?

• What size sample should be obtained if he wishes the estimate to be within 2 points with 99% confidence?

  • To estimate the population mean with a margin of error E at a $(1-\alpha) \cdot 100$ level of confidence:

  • $n = ( \frac{z_{\frac{\alpha}{2}} \cdot s}{E})^2$ rounded up to the next integer when no prior estimate of $p$ is available.

  • $n = ( \frac{2.575 \cdot 15.9}{2})^2 = 420$

• What size sample should be obtained if he wishes the estimate to be within 2 points with 95% confidence?

  • To estimate the population mean with a margin of error E at a $(1-\alpha) \cdot 100$ level of confidence:

  • $n = ( \frac{z_{\frac{\alpha}{2}} \cdot s}{E})^2$ rounded up to the next integer when no prior estimate of $p$ is available.

  • $n = ( \frac{1.96 \cdot 15.9}{2})^2 = 243$

• How does the decrease in confidence affect the sample size required?

  • Decreasing the confidence level decrease the sample size needed.