Determining Sample Size. Assume that each sample is a simple rand...

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

Statistics

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

2:39 minutes

Problem 7.3.18

Textbook 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?

Verified step by step guidance

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Size Determination

Sample size determination is a statistical process used to calculate the number of observations needed in a sample to achieve a desired level of precision and confidence in estimating a population parameter. It involves considering factors such as the population size, the expected variability in the data, and the desired confidence level, which in this case is 95%.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence. In this scenario, a 95% confidence interval means that if we were to take many samples and build intervals, approximately 95% of those intervals would contain the true population mean of diastolic blood pressures.

Standard Deviation and Variability

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of this question, understanding the population standard deviation (σ) is crucial for determining how much the sample standard deviation (s) can deviate from it, specifically within the 1% margin of error, which impacts the required sample size for accurate estimation.

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Key Concepts

Sample Size Determination

Confidence Interval

Standard Deviation and Variability

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