Question

For each of the following three sample sizes, construct the \(95 \%\) confidence interval for the population proportion. 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

Increasing sample size decreases the interval width because standard error diminishes, leading to more precise estimates.

Step-by-step solution

- Understanding Confidence Interval Formula

The formula for a confidence interval for a population proportion is given by \[ \text{CI} = \text{p} \, \pm \, z \sqrt{\frac{\text{p}(1-\text{p})}{N}} \] where \(\text{p}\) is the sample proportion, \(z\) is the z-score corresponding to the confidence level, and \(N\) is the sample size. For a 95% confidence level, the z-score (\(z\)) is 1.96.

- Calculate Standard Error for Sample A (N = 100)

Calculate the standard error for the sample size \(N = 100\) using the formula \( \sqrt{\frac{\text{p}(1-\text{p})}{N}} \). Substituting the values, we get \[ \text{SE} = \sqrt{\frac{0.40 \times 0.60}{100}} = \sqrt{0.0024} \approx 0.049 \]

- Calculate Confidence Interval for Sample A

The confidence interval for Sample A can be calculated using the formula: \[ 0.40 \pm 1.96 \times 0.049 \] This results in the interval \[ 0.40 \pm 0.096 \] Therefore, the confidence interval is approximately \[ (0.304, 0.496) \]

- Calculate Standard Error for Sample B (N = 1000)

Calculate the standard error for the sample size \(N = 1000\). \[ \text{SE} = \sqrt{\frac{0.40 \times 0.60}{1000}} = \sqrt{0.00024} \approx 0.0155 \]

- Calculate Confidence Interval for Sample B

The confidence interval for Sample B can be calculated using the formula: \[ 0.40 \pm 1.96 \times 0.0155 \] This results in the interval \[ 0.40 \pm 0.0304 \] Therefore, the confidence interval is approximately \[ (0.3696, 0.4304) \]

- Calculate Standard Error for Sample C (N=10,000)

Calculate the standard error for the sample size \(N = 10,000\). \[ \text{SE} = \sqrt{\frac{0.40 \times 0.60}{10000}} = \sqrt{0.000024} \approx 0.0049 \]

- Calculate Confidence Interval for Sample C

The confidence interval for Sample C can be calculated using the formula: \[ 0.40 \pm 1.96 \times 0.0049 \] This results in the interval \[ 0.40 \pm 0.0096 \] Therefore, the confidence interval is approximately \[ (0.3904, 0.4096) \]

- Analyze the Result

As the sample size increases, the width of the confidence interval decreases. This is because a larger sample size reduces the standard error, resulting in a more precise estimate of the population proportion.

Key concepts

population proportion

The population proportion represents the fraction of the total population that has a particular characteristic. For example, if 40% of a city’s population supports a certain policy, the population proportion is 0.40. In sample studies, this proportion is crucial because it helps in estimating what the true population proportion might be based on the sample data collected. When dealing with confidence intervals, we use the sample proportion as an estimate of the population proportion to calculate the range in which the true population value lies. This is represented by the symbol \(\text{p}\). The precision of this estimate depends heavily on the sample size and the variability within the population.

sample size

Sample size is the number of observations in a sample and is denoted by the symbol \(N\). In the problem, we have three sample sizes: 100, 1000, and 10,000. The importance of sample size lies in its impact on the reliability of the estimate. Larger sample sizes tend to produce more accurate estimates of the population parameters. This occurs because larger samples better capture the variability within the population. With the increase in sample size, the standard error decreases, leading to narrower confidence intervals. This implies that our estimate of the population proportion becomes more precise. Consequently, understanding and choosing an appropriate sample size is a critical aspect of designing studies and experiments.

standard error

The standard error measures how much the sample proportion is expected to vary from the true population proportion. It is calculated using the formula \[ \text{SE} = \sqrt{\frac{\text{p}(1-\text{p})}{N}} \], where \(\text{p}\) is the sample proportion, and \(N\) is the sample size. In this exercise, as we increase the sample size from 100 to 10,000, we observe that the standard error decreases. For simplicity, you can think of standard error as the

confidence level

The confidence level defines how certain we are that the confidence interval contains the true population proportion. Typically expressed as a percentage (e.g., 95%), it indicates the likelihood that the interval will include the true proportion if we were to take many samples. The higher the confidence level, the wider the confidence interval, since we are accounting for more uncertainty. For a 95% confidence level, we use a z-score of 1.96. This z-score is derived from the standard normal distribution and corresponds to the point which includes 95% of the data. In practical terms, a 95% confidence level means that if we were to repeat the study 100 times, the true population proportion would fall within our interval 95 times out of those 100.

z-score

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is a measure of how many standard deviations an element is from the mean. In the context of confidence intervals, the z-score is used to calculate the margin of error. For a 95% confidence level, the z-score is 1.96. This means that we expect the sample proportion to lie within 1.96 standard errors of the true population proportion. Essentially, the z-score bridges the gap between the sample data and the normal distribution, helping us estimate where the true population parameter lies. The exact z-score value depends on the desired confidence level; higher confidence levels correspond to higher z-scores, reflecting greater certainty and wider intervals.