Question

The accompanying data represent the total travel tax (in dollars) for a 3-day business trip in 8 randomly selected cities. A normal probability plot suggests the data could come from a population that is normally distributed. A boxplot indicates there are no outliers. Complete parts (a) through (c) below. (a) Determine a point estimate for the population mean travel tax. A point estimate for the population mean travel tax is $ (Round to two decimal places as needed.) (b) Construct and interpret a 90% confidence interval for the mean tax paid for a three-day business trip. Select the correct choice below and fill in the answer boxes to complete your choice. (Round to two decimal places as needed.) A. There is a % probability that the mean travel tax for all cities is between $ and $ B. The travel tax is between $ and $ for % of all cities. C. One can be % confident that the mean travel tax for all cities is between $ and $ D. One can be % confident that all cities have a travel tax between $ and $ (c) What would you recommend to a researcher who wants to increase the precision of the interval, but does not have access to additional data?

Answer 1

A point estimate for the population mean is calculated from sample data. A 90% confidence interval for the mean travel tax can be constructed using the sample mean, standard deviation, and the critical value from the standard normal distribution. To increase precision without more data, a few options are available but all come with trade-offs.

Step-by-step explanation:

Understanding Confidence Intervals

Part (a): Estimating the Population Mean Travel Tax

To find a point estimate for the population mean travel tax, we calculate the sample mean by summing all the individual travel taxes for the 8 cities and dividing by the number of cities. This provides us with the best estimate of the average travel tax across all cities.

Part (b): Calculating and Interpreting a 90% Confidence Interval

For constructing a 90% confidence interval for the mean travel tax, we assume that the sample mean is normally distributed due to the sample size and the data being normally distributed. We use the critical value from the standard normal distribution corresponding to a 90% confidence level and the standard error of the mean to compute the margin of error. By adding and subtracting this margin from the sample mean, we get the confidence interval. The correct interpretation would be option A or C, indicating that we can be 90% confident that the true mean travel tax for all cities lies within this interval.

Part (c): Recommendations to Increase Precision

To increase the precision of the interval without more data, one approach would be to increase the confidence level. However, this would actually make the interval wider. Instead, a researcher could reduce the confidence level, which would result in a narrower interval, but this would reduce the confidence we have that the interval contains the population mean. Without the ability to collect more data to increase the sample size, there are inherent limitations to how much the precision of the confidence interval can be improved.