Use the given sample data and confidence level. In each...

Use the given sample data and confidence level. In each case, (a) find the best point estimate of the population proportion $p ;(b)$ identify the value of the margin of error $E ;(c)$ construct the confidence interval; (d) write a statement that correctly interprets the confidence interval. In a study of cell phone use and brain hemispheric dominance, an Intemet survey was e-mailed to 5000 subjects randomly selected from an online group involved with ears. 717 surveys were returned. Construct a $90 \%$ confidence interval for the proportion of retumed surveys.

Use the given sample data and confidence level. In each case, (a) find the best point estimate of the population proportion $p ;(b)$ identify the value of the margin of error $E ;(c)$ construct the confidence interval; (d) write a statement that correctly interprets the confidence interval.
In a study of cell phone use and brain hemispheric dominance, an Intemet survey was e-mailed to 5000 subjects randomly selected from an online group involved with ears. 717 surveys were returned. Construct a $90 \%$ confidence interval for the proportion of retumed surveys.

Key Concepts

Sample Proportion (Point Estimate)

The sample proportion is an estimate of the population proportion based on the data from a sample. It is calculated by dividing the number of positive outcomes (or successes) by the total number of observations in the sample. In the context of confidence intervals, this sample proportion serves as the best point estimate for the true but unknown proportion in the entire population.

Margin of Error

The margin of error quantifies the amount of random sampling error in the estimate of the population parameter. It is derived using the critical value corresponding to the desired level of confidence (such as from the Z-distribution for large samples) and the standard error of the sample proportion. This margin is added and subtracted from the point estimate to create an interval that is likely to contain the true proportion.

Confidence Interval

A confidence interval provides a range of plausible values for the population proportion. It is calculated by taking the point estimate and adding and subtracting the margin of error. This interval is constructed under the assumption that the sampling distribution of the estimate approximates a normal distribution, especially with large sample sizes, and reflects the chosen level of confidence.

Interpretation of Confidence Interval

Interpreting a confidence interval involves explaining that if the same sampling procedure were repeated many times, the interval constructed in this manner would contain the true population proportion a specified percentage of the time (for example, 90% of intervals would capture the parameter). This does not imply that there is a 90% probability that the particular interval calculated from one sample contains the true parameter, but rather reflects the long-run performance of the method.

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