"InfidelityAccording to menstuff.org, 22% of married men have “strayed” at least once during their married lives.


b. A survey of 500 married men indicated that 122 have “strayed” at least once during their married life. Construct a 95% confidence interval for the population proportion of married men who have strayed. Use this interval to assess the accuracy of the statement made by menstuff.org.


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Verified step by step guidance

1. Identify the sample proportion \( \hat{p} \) by dividing the number of men who have strayed by the total number surveyed. That is, calculate \( \hat{p} = \frac{122}{500} \).
2. Determine the confidence level, which is 95%, and find the corresponding critical value \( z^* \) from the standard normal distribution. For 95%, this value is typically around 1.96.
3. Calculate the standard error (SE) of the sample proportion using the formula: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \] where \( n = 500 \) is the sample size.
4. Construct the confidence interval using the formula: \[ \hat{p} \pm z^* \times SE \] This will give you the lower and upper bounds of the 95% confidence interval for the population proportion.
5. Interpret the confidence interval by checking if the value 0.22 (the proportion stated by menstuff.org) lies within the interval. If it does, the survey data supports the claim; if not, it suggests a difference.

Key Concepts

Population Proportion

The population proportion represents the fraction of individuals in a population that have a particular characteristic, such as having 'strayed' in this context. It is often denoted by p and is estimated using sample data to make inferences about the entire population.

Confidence Interval for a Proportion

A confidence interval provides a range of plausible values for a population proportion based on sample data. At a 95% confidence level, it means we are 95% confident that the true population proportion lies within this interval, accounting for sampling variability.

Hypothesis Testing and Interpretation

Using the confidence interval, we can assess the accuracy of a claimed population proportion by checking if the claimed value lies within the interval. If it does, the claim is plausible; if not, the claim may be inaccurate or require further investigation.