Question

In Exercises 6.11 to 6.14, use the normal distribution to find a confidence interval for a proportion \(p\) given the relevant sample results. Give the best point estimate for \(p,\) the margin of error, and the confidence interval. Assume the results come from a random sample. A \(95 \%\) confidence interval for \(p\) given that \(\hat{p}=\) 0.38 and \(n=500\)

Short answer

The best point estimate for \(p\) is 0.38, the margin of error is 0.0437 and the 95% confidence interval for \(p\) is [0.3363, 0.4237]

Step-by-step solution

Point Estimate

The best point estimate for the population proportion \(p\) is the sample proportion \(\hat{p}\). Therefore, the best point estimate for \(p\) is 0.38 in this case.

Calculate Standard Error

We calculate the standard error of the proportion (SEP) using the formula SEP = \(\sqrt{ \hat{p}(1-\hat{p}) / n}\) where \(n\) is the sample size, and \(\hat{p}\) is the sample proportion. Substituting the given values, SEP = \(\sqrt{0.38(1-0.38)/500} = 0.0223\)

Margin of Error

Next, we calculate the margin of error using the formula: Margin of Error = Z * SEP. For a 95% confidence interval, the z-score is 1.96 (from z-table). Hence, Margin of Error = 1.96 * 0.0223 = 0.0437

Confidence Interval

Finally, we calculate the confidence interval using the point estimate and the margin of error. The confidence interval = \(\hat{p}\pm\) Margin of Error. Hence, Confidence interval = [0.38 - 0.0437, 0.38 + 0.0437] = [0.3363, 0.4237]

Key concepts

Normal Distribution

When we talk about the confidence interval for a proportion, we are often considering the normal distribution as the basis for our calculations. Normal distribution, also known as the Gaussian distribution, is a symmetric, bell-shaped distribution where most of the observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions.

It’s important to note that the normal distribution is used in the context of confidence intervals when certain conditions are met. Typically, these conditions require a sufficiently large sample size (usually a rule of thumb is n > 30) such that the sampling distribution of the proportion can be approximated as normal based on the Central Limit Theorem, regardless of the distribution of the population from which the sample was drawn.

Margin of Error

The margin of error is crucial to understanding the range within which we expect the population proportion to lie, given a certain level of confidence. It reflects how much we can expect our estimate to vary if we were to take many samples. A smaller margin of error indicates a more precise estimate.

The formula for the margin of error includes the standard error and a multiplier derived from the z-score: Margin of Error = Z * SEP. Essentially, it adjusts the width of the confidence interval. For a 95% confidence level, common in statistical analysis, the z-score is 1.96, representing the standard deviation limits on either side of a normal distribution curve that contains 95% of the data.

Standard Error of the Proportion

The standard error of the proportion (SEP) quantifies how much we would expect the sample proportion to vary from one random sample to another. It is a pivotal component in calculating the margin of error and, subsequently, the confidence interval.

The SEP is calculated using the formula \( SEP = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \) where \( \hat{p} \) is the sample proportion and \(n\) is the sample size. A key point to remember is that the SEP decreases as the sample size increases, indicating that larger samples give us more precise estimates of the population proportion.

Z-Score

The z-score is a statistical measure that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations. In the context of confidence intervals, z-scores are used to determine the margin of error by indicating how many standard deviations away from the point estimate the interval extends.

For different confidence levels, there are corresponding z-scores: 1.96 for 95%, 2.58 for 99%, and so on. These scores are critical when working with the normal distribution, as they help us understand the spread and probability of the data within a given range in relation to the mean.

Point Estimate

A point estimate provides an actual value as an estimate of a population parameter based on sample data. When we calculate a confidence interval for a proportion, our point estimate is the sample proportion, often denoted as \( \hat{p} \).

The point estimate is the starting center of the confidence interval, and by itself, it does not convey information about the precision or reliability of the estimate; that's why we use it with the margin of error to construct the confidence interval. The interval thus signifies a range within which we believe the true population proportion may fall, with a certain level of confidence.