Question

For each of the following three sample sizes, construct the \(95 \%\) confidence interval. Use a sample proportion of \(0.40\) throughout. What happens to interval width as sample size increases? Why? $$ \begin{aligned} P_{s} &=0.40 \\ \text { Sample A: } N &=100 \\ \text { Sample B: } N &=1000 \\ \text { Sample C: } N &=10,000 \end{aligned} $$

Short answer

The confidence interval width decreases as sample size increases because the standard error decreases with larger sample sizes.

Step-by-step solution

Identify the given parameters

The sample proportion, denoted as \(P_s\), is given as 0.40. The confidence level is 95%, which corresponds to a Z-score (\

Key concepts

Sample Size

The term 'sample size' refers to the number of observations or data points collected in a study. It is denoted by the letter N. The importance of sample size in statistics cannot be overstated, as it directly affects the accuracy and reliability of the results. A larger sample size provides more information and tends to produce more reliable estimates.
In the context of our exercise, we have three different sample sizes: N = 100, N = 1000, and N = 10,000.

• Sample A: N = 100
• Sample B: N = 1000
• Sample C: N = 10,000

As the sample size increases from 100 to 10,000, the confidence intervals will become narrower because larger sample sizes reduce the standard error.

Proportion

The term 'proportion' represents the fraction of the total that possesses a particular characteristic of interest. It is often denoted as P or \(P_s\). In our problem, the sample proportion (\(P_s\)) is given as 0.40.

This means in our sample, 40% of the observations possess the trait we are analyzing. Calculating the confidence interval around this proportion will allow us to understand the range within which the true population proportion is likely to fall.

• Proportion = Number of favorable outcomes / Total number of outcomes
• \(P_s = 0.40\)

Z-score

The Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is expressed in terms of standard deviations from the mean. In the context of confidence intervals, the Z-score is used to determine the margin of error.

For a 95% confidence level, the Z-score is approximately 1.96. This value comes from standard normal distribution tables and signifies that 95% of the data falls within 1.96 standard deviations of the mean.

• Z-score for 95% confidence level = 1.96

Calculating the Z-score helps us to adjust our interval to the desired confidence level, ensuring that our interval estimate is accurate.

Interval Width

The interval width in a confidence interval is the range between the lower and upper bounds. It is influenced by both the sample size and the standard error. A narrower interval width indicates more precise estimates.

As shown in the exercise, as the sample size increases from 100 to 10,000, the interval width decreases. This happens because the standard error, which is inversely proportional to the square root of the sample size, gets smaller.

The interval width is given by the formula: \[ \text{Interval Width} = 2 \times (Z \times \text{Standard Error}) \]

• Sample A: Wider interval
• Sample B: Narrower interval
• Sample C: Narrowest interval

Increasing the sample size improves the precision of our confidence interval, thereby reducing the interval width.