According to Franchise Business Review, over 50 % of all...

According to Franchise Business Review, over $50 \%$ of all food franchises earn a profit of less than $$\$ 50,000$$ a year. In the sample of 142 casual dining restaurants contained in the file CasualDining, 81 earned a profit of less than $$\$ 50,000$$ last year. a. What is the point estimate of the proportion of casual dining restaurants that earned a profit of less than $$\$ 50,000$$ last year? b. Determine the margin of error and provide a $95 \%$ confidence interval for the proportion of casual dining restaurants that earned a profit of less than $$\$ 50,0004$ last year. c. How large a sample is needed if the desired margin of error is .03?

According to Franchise Business Review, over $50 \%$ of all food franchises earn a profit of less than $$\$ 50,000$$ a year. In the sample of 142 casual dining restaurants contained in the file CasualDining, 81 earned a profit of less than $$\$ 50,000$$ last year.
a. What is the point estimate of the proportion of casual dining restaurants that earned a profit of less than $$\$ 50,000$$ last year?
b. Determine the margin of error and provide a $95 \%$ confidence interval for the proportion of casual dining restaurants that earned a profit of less than $$\$ 50,0004$ last year.
c. How large a sample is needed if the desired margin of error is .03?

Key Concepts

Point Estimate

A point estimate is a single value that serves as a best guess or approximation for an unknown parameter in the population. In the context of proportions, the point estimate is calculated as the sample proportion, which provides an immediate estimate of the true proportion of success or occurrence in the population.

Confidence Interval

A confidence interval is a range of values, derived from the sample data, that is believed to contain the true population parameter with a certain level of confidence (e.g., 95%). For proportions, this interval is constructed by taking the point estimate and adding and subtracting the product of the corresponding critical value from the standard normal distribution and the standard error of the estimate, thereby quantifying the uncertainty associated with the estimation process.

Margin of Error

The margin of error is the maximum expected difference between the true population parameter and the point estimate due to sampling variability. It is determined by multiplying the standard error by the critical value (such as the z-score for a given confidence level). In the context of a confidence interval for proportions, the margin of error provides a measure of the precision of the estimated proportion.

Sample Size Determination

Sample size determination is a process used to calculate the minimum number of observations required to achieve a desired level of precision for an estimate. For proportion estimates, the desired margin of error can be set, and then the necessary sample size is computed using the formula that relates the margin of error, the estimated proportion, and the appropriate critical value from the standard normal distribution. This process ensures that the study or survey is sufficiently powered to provide a confidence interval with the specified width.

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