We have given the number of successes and the sample size...

We have given the number of successes and the sample size for a simple random sample from a population. In each case, do the following tasks. a. Determine the sample proportion. b. Decide whether using the one-proportion z-interval procedure is appropriate. c. If appropriate, use the one-proportion z-interval procedure to find the confidence interval at the specified confidence level. $$x=10, n=40,90 \% \text { level. }$$

We have given the number of successes and the sample size for a simple random sample from
a population. In each case, do the following tasks.
a. Determine the sample proportion.
b. Decide whether using the one-proportion z-interval procedure is appropriate.
c. If appropriate, use the one-proportion z-interval procedure to find the confidence interval at the specified confidence level.
$$x=10, n=40,90 \% \text { level. }$$

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Key Concepts

Sample Proportion

The sample proportion is a point estimate of the true proportion of successes in the population, calculated by dividing the number of successes by the total sample size. It provides a simple measure of the observed success rate and is used as a basis for further inference about the population parameter.

One-Proportion z-Interval Procedure

This procedure is used to construct a confidence interval for a population proportion when the sample size is sufficiently large. It relies on the normal approximation to the binomial distribution and uses the z-distribution to account for variability in the estimate.

Confidence Interval

A confidence interval is an interval estimate, computed from the sample data, within which the true population parameter is expected to fall with a specified probability (confidence level). It quantifies the uncertainty associated with the sample estimate.

Conditions for Normal Approximation

For the one-proportion z-interval procedure to be appropriate, the sample must satisfy certain conditions such as having a large enough sample size so that both the number of successes and failures are above a minimum threshold (often at least 10). This ensures the sampling distribution of the sample proportion is approximately normal, justifying the use of the z-distribution.

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