The average cost per night of a hotel room in New York City...

The average cost per night of a hotel room in New York City is $\$ 273$ (SmartMoney, March 2009 ). Assume this estimate is based on a sample of 45 hotels and that the sample standard deviation is $\$ 65 .$ $$ \begin{array}{l}{\text { a. With } 95 \% \text { confidence, what is the margin of error? }} \\ {\text { b. What is the } 95 \% \text { confidence interval estimate of the population mean? }} \\ {\text { c. Two years ago the average cost of a hotel room in New York City was } \$ 229 . \text { Discuss }} \\ {\text { the change in cost over the two-year period. }}\end{array} $$

The average cost per night of a hotel room in New York City is $\$ 273$ (SmartMoney, March
2009 ). Assume this estimate is based on a sample of 45 hotels and that the sample standard
deviation is $\$ 65 .$
$$
\begin{array}{l}{\text { a. With } 95 \% \text { confidence, what is the margin of error? }} \\ {\text { b. What is the } 95 \% \text { confidence interval estimate of the population mean? }} \\ {\text { c. Two years ago the average cost of a hotel room in New York City was } \$ 229 . \text { Discuss }} \\ {\text { the change in cost over the two-year period. }}\end{array}
$$

Key Concepts

Comparing Population Means Over Time

This concept involves using statistical estimates from different time periods to assess changes in the population parameter. By comparing confidence intervals or conducting hypothesis tests, one can determine whether observed differences between periods are statistically significant or could be attributed to sampling variability.

t-Distribution

The t-distribution is used in constructing confidence intervals for the population mean when the sample size is small and the population standard deviation is unknown. It adjusts for extra uncertainty by having heavier tails than the normal distribution, and the critical values depend on the degrees of freedom, which are based on the sample size.

Margin of Error

Margin of error represents the maximum amount by which the sample estimate is expected to differ from the true population parameter. It is calculated by multiplying the critical value (from the t or z distribution, depending on the sample size and whether the population standard deviation is known) by the standard error of the sample statistic. This concept is central in estimating the precision of a confidence interval.

Confidence Interval

A confidence interval is a range derived from the sample data within which the true population parameter is expected to lie with a given level of confidence. It combines the sample mean with the margin of error to provide a probabilistic assessment that the interval captures the true mean, thereby reflecting the reliability of the estimate.

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