Question

A random sample of \(n=300\) observations from a binomial population produced \(x=263\) successes. Find a \(90 \%\) confidence interval for \(p\) and interpret the interval.

Short answer

Answer: We can be 90% confident that the true proportion of success, p, in the population lies between 0.8486 and 0.9048.

Step-by-step solution

Find the point estimate of the proportion p

Since we have \(n=300\) observations with \(x=263\) successes, the point estimate of the proportion of success, \(p\), is given by $$\hat{p} = \frac{x}{n} = \frac{263}{300}.$$ Calculate the value of \(\hat{p}\): $$\hat{p} \approx 0.8767.$$

Calculate the standard error of the estimate

The standard error of the estimate, \(SE_{\hat{p}}\), is given by the formula: $$SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.$$ Substitute the values of \(\hat{p}\) and \(n\) into the formula: $$SE_{\hat{p}} \approx \sqrt{\frac{0.8767 \times 0.1233}{300}}.$$ Calculate the value of \(SE_{\hat{p}}\): $$SE_{\hat{p}} \approx 0.0171.$$

Find the critical value for a 90% confidence interval

We are given a confidence level of \(90\%\). Since the distribution is binomial, we can use a normal (Z) distribution to approximate the critical value. The two tail critical Z value for a 90% confidence interval is \(1.645\) (You can find these values in Z-distribution tables or use statistical calculators available online).

Calculate the margin of error

To find the margin of error, we multiply the standard error of the estimate, \(SE_{\hat{p}}\), by the critical Z-value: $$\text{Margin of Error} = Z_{\alpha/2} \times SE_{\hat{p}}.$$ Substitute the values of \(Z_{\alpha/2}\) and \(SE_{\hat{p}}\) into the formula: $$\text{Margin of Error} \approx 1.645 \times 0.0171.$$ Calculate the value of the Margin of Error: $$\text{Margin of Error} \approx 0.0281.$$

Find the confidence interval

Use the point estimate and margin of error to find the 90% confidence interval for \(p\): $$(\hat{p}-\text{Margin of Error},\; \hat{p}+\text{Margin of Error}) = (0.8767 - 0.0281,\; 0.8767 + 0.0281) = (0.8486,\; 0.9048).$$

Interpret the interval

The 90% confidence interval suggests that we can be \(90\%\) confident that the true proportion of success \(p\) in the population lies between 0.8486 and 0.9048.

Key concepts

Binomial Distribution

A binomial distribution is a type of probability distribution. It describes the outcome of a sequence of experiments where each experiment, known as a trial, has exactly two possible outcomes: success or failure. For a trial to be classified as binomial, these conditions need to be met:

• Each trial is independent.
• The probability of success, denoted as \(p\), remains the same for every trial.
• There is a fixed number of trials, \(n\).

In the exercise, we are given that there are 300 trials (observations) and 263 successes. Using this information, we can analyze the properties of the binomial distribution to find a confidence interval, which helps us understand more about the population from which the sample was drawn.

Point Estimate

The point estimate is a single value that approximates a population parameter. In our situation, it relates to estimating the probability of success \(p\) in the binomial distribution using sample data.
The point estimate of the proportion \(\hat{p}\) is determined by the formula:\[\hat{p} = \frac{x}{n}\]where \(x\) is the number of successes, and \(n\) is the total number of trials. For our exercise, plugging in the values, we get:\[\hat{p} = \frac{263}{300} \approx 0.8767\]
This means we estimate that roughly 87.67% of the population exhibits the successful trait observed.

Standard Error

The standard error measures the accuracy of the point estimate by quantifying the variability of the estimate in repeated samples from the population. For a proportion, the standard error \(SE_{\hat{p}}\) is calculated as follows:
\[SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]
For the given exercise, with \(\hat{p} \approx 0.8767\) and \(n = 300\), we calculate the standard error as:\[SE_{\hat{p}} \approx \sqrt{\frac{0.8767 \times 0.1233}{300}} \approx 0.0171\]
A smaller standard error suggests that our sample mean is a precise estimate of the population mean, implying reliability in our predictions.

Margin of Error

The margin of error quantifies the uncertainty surrounding a point estimate and provides a range within which the true population parameter is expected to fall.
It is calculated by multiplying the standard error by the critical value from the Z-distribution that corresponds to the desired confidence level. For a 90% confidence interval, the critical value \(Z_{\alpha/2}\) is approximately 1.645.
Using the standard error from our exercise:\[\text{Margin of Error} = 1.645 \times 0.0171 \approx 0.0281\]
This value tells us that we can be reasonably certain that the actual population proportion lies within ±0.0281 of our point estimate, indicating the level of confidence in our findings.