Question

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 the proportion of the population in Category A given that \(23 \%\) of a sample of 400 are in Category \(\mathrm{A}\)

Short answer

The best point estimate for p is 0.23, the margin of error is 0.0406 and the 95% confidence interval for the proportion is (0.1894, 0.2706)

Step-by-step solution

Determine the point estimate

The point estimate is the proportion of the sample that falls into the given category, in this case, category A. We have that 23% of the sample, or \( 0.23 \), is in Category A. So, our point estimate \( p \) is \( 0.23 \)

Determine the Z-score

The Z-score refers to how many standard deviations an element is from the mean. This will tell us the level of confidence we need. For a 95% confidence interval, the Z-score is \( 1.96 \)

Calculate the standard error

The standard error (SE) can be calculated using the formula: \( SE = \sqrt{p(1-p) / n}\), where \( p = 0.23 \) is the proportion and \( n = 400 \) is the sample size. Substituting these values into the formula gives \( SE = \sqrt{0.23 * (1 - 0.23) / 400} = 0.0207 \)

Calculate the Margin of Error (ME)

The margin of error is calculated by multiplying the Z-score by the standard error. Thus, \( ME = Z * SE = 1.96 * 0.0207 = 0.0406. \)

Establish the confidence interval

We will calculate the confidence interval as point estimate ± margin of error, which simplifies to, \( 0.23 ± 0.0406 \). Thus the 95% confidence interval for the proportion is \( (0.1894, 0.2706) \)

Key concepts

Normal Distribution

When calculating confidence intervals for proportions, the normal distribution, sometimes called the Gaussian distribution, plays a vital role. This is a continuous probability distribution symmetrically shaped like a bell, where most outcomes cluster around the mean or center, with probabilities tapering off as you move away.
Imagine throwing a dart numerous times at a target. Most of your throws are likely to hit near the bullseye, with fewer scatters toward the edges of the board. In statistics, if a large number of sample proportions are calculated from samples of the same size, they tend to form a normal distribution around the true population proportion.
This property allows statisticians to make probabilistic statements about the population proportion based on the sample proportion. The normal distribution is essential for constructing confidence intervals because it provides a basis for estimating the likelihood that any given sample proportion falls within a certain distance from the true population proportion.

Point Estimate

A point estimate serves as the best single guess or value for a population parameter based on sample data. In your exercise, the point estimate is the sample proportion of Category A members, which is 23% or 0.23. This figure represents our best estimate of the true proportion of the population in Category A.
The point estimate by itself doesn't say much about the reliability or precision of the estimate. That's where the confidence interval comes into play, enhancing the point estimate by providing a range that the true population parameter is likely to fall within. Understanding the notion of a point estimate is crucial because it is the starting point for calculating the confidence interval and gives us a central value around which the interval is constructed.

Z-score

In statistics, the Z-score is a measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. For confidence intervals, the Z-score is used to determine how 'confident' we want to be in our estimate.
In the example, a Z-score of 1.96 corresponds to a 95% confidence level. This means that if we were to take many samples and calculate confidence intervals using this Z-score, 95% of those intervals would contain the true population proportion. Essentially, the Z-score helps quantify the level of certainty regarding where the population parameter lies in relation to our sample estimate.

Standard Error

The standard error (SE) measures how much the sample proportion is expected to vary from one sample to another. It's a type of 'standard deviation' for sampling distribution. The smaller the standard error, the more precise the estimate is likely to be.
In the provided solution, the standard error is calculated using the point estimate of the sample proportion and the size of the sample. This calculation acknowledges the statistical truth that there's variability every time we draw a sample from the population. Knowing the standard error is a pivotal step in calculating the margin of error, which ultimately determines the width of the confidence interval.

Margin of Error

The margin of error (ME) encapsulates the range of error in a point estimate, representing the furthest we reasonably believe our estimate could be from the true population parameter. It’s influenced by the size of the sample and how confident we want to be in our point estimate.
In your exercise, the margin of error is found by multiplying the Z-score by the standard error, incorporating both our desired confidence level and the variability of the estimate. The presence of the margin of error in reporting survey results, such as a confidence interval, is crucial because it communicates the precision of the sample proportion as an estimate of the population proportion. It effectively indicates the 'buffer zone' we apply to our point estimate.