Question

Determining Sample Size. Assume that each sample is a simple random sample obtained from a normally distributed population.

You want to estimate for the population of diastolic blood pressures of air traffic controllers in the United States. Find the minimum sample size needed to be 95% confident that the sample standard deviation s is within 1% of σ. Is this sample size practical?

Accepted Answer

Step 1: Understand the problem. The goal is to determine the minimum sample size (n) required to estimate the population standard deviation (σ) with 95% confidence, such that the sample standard deviation (s) is within 1% of σ. This involves using the chi-square distribution, as it is commonly used for variance and standard deviation estimation. Step 2: Recall the formula for the confidence interval of the population standard deviation (σ). The formula is derived from the chi-square distribution: $$ \frac{(n-1)s^2}{\chi^2_{\alpha/2}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}} $$, where $$ \chi^2_{\alpha/2} $$ and $$ \chi^2_{1-\alpha/2} $$ are the critical values of the chi-square distribution for a given confidence level (1 - α). Step 3: Set up the margin of error condition. The problem specifies that the sample standard deviation (s) must be within 1% of the population standard deviation (σ). This means the margin of error (E) is $$ 0.01\sigma . $$Using the chi-square formula, the margin of error can be expressed as $$ E = \sigma \sqrt{\frac{\chi^2_{1-\alpha/2} - \chi^2_{\alpha/2}}{(n-1)\chi^2_{\alpha/2}\chi^2_{1-\alpha/2}}} . $$Step 4: Rearrange the formula to solve for the sample size (n). To find the minimum sample size, isolate $$ n  in $$the inequality. This requires algebraic manipulation of the chi-square formula and substituting the given values: confidence level (95%, so $$ \alpha = 0.05 $$), and the margin of error (1% of σ). Use chi-square critical values corresponding to $$ \alpha/2 = 0.025 $$ and $$ 1-\alpha/2 = 0.975 . $$Step 5: Evaluate the practicality of the sample size. Once the minimum sample size (n) is calculated, consider whether it is feasible to collect this many samples in practice. Factors such as time, cost, and accessibility of air traffic controllers should be taken into account to determine if the sample size is practical.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Sample Size Determination

Confidence Interval

Standard Deviation and Variability