Question

Information about a sample is given. Assume that the sampling distribution is symmetric and bell-shaped. P^ = .33 and the standard error is 0.05.

Use the information to give a 95% confidence interval. Round your answer to two decimal places.
The 95% confidence interval is _____ to ______.

Answer 1

The 95% confidence interval is 0.23 to 0.43, indicating that we can be 95% confident the true population proportion lies within this range.

Step-by-step explanation:

A 95% confidence interval is calculated using the formula:
`\(\bar{p} \pm Z * SE\)` , where
`\(\bar{p}\)` is the sample proportion, Z is the Z-score corresponding to the desired confidence level, and SE is the standard error. For a 95% confidence interval, the Z-score is approximately 1.96.

Given
`\(\bar{p} = 0.33\) and \(SE = 0.05\),` the calculation is as follows:

`\[0.33 \pm 1.96 * 0.05\]`

`\[0.33 \pm 0.098\]`

This yields the interval [0.23, 0.43] when rounded to two decimal places.

In the context of a confidence interval, the result suggests that we can be 95% confident that the true population proportion (p) lies between 0.23 and 0.43. This range provides a level of precision for the estimate based on the sample data.

It's important to note that the width of the confidence interval is influenced by the standard error. A smaller standard error results in a narrower interval, indicating a more precise estimate. The Z-score accounts for the desired confidence level and reflects the standard normal distribution's critical values. In this case, the symmetric and bell-shaped assumption aligns with the properties of a standard normal distribution.

Answer 2

The 95% confidence interval is 0.23 to 0.43.

Step-by-step explanation:

In statistics, a 95% confidence interval provides a range of values within which we can reasonably expect the true population parameter to lie. Given that the sampling distribution is symmetric and bell-shaped, and the point estimate
`(\( \hat{p} \))` is 0.33 with a standard error of 0.05, we can construct the confidence interval.

The formula for the confidence interval for a proportion p is given by
`\( \hat{p} \pm z * SE \)`, where z is the z-score corresponding to the desired level of confidence. For a 95% confidence interval, z is approximately 1.96.

So, the lower bound of the interval is 0.33 - (1.96 x 0.05) = 0.23 , and the upper bound is 0.33 + (1.96 x 0.05) = 0.43. Therefore, the 95% confidence interval is 0.23 to 0.43.

This means that if we were to take many samples and construct a confidence interval for the true proportion in each sample, we would expect about 95% of those intervals to contain the true population proportion. The narrow width of this interval (0.20) indicates relatively high precision in estimating the true proportion, demonstrating the confidence in the precision of our estimate.