Question

One of the major costs involved in planning a summer vacation is the cost of lodging. Even within a particular chain of hotels, costs can vary substantially depending on the type of room and the amenities offered. \(^{4}\) Suppose that we randomly select 50 billing statements from each of the computer databases of the Marriott, Radisson, and Wyndham hotel chains, and record the nightly room rates. $$\begin{array}{lccc} & \text { Marriott } & \text { Radisson } & \text { Wyndham } \\\\\hline \text { Sample average } & \$ 170 & \$ 145 & \$ 150 \\\\\text { Sample standard deviation } & 17.5 & 10 & 16.5\end{array}$$ a. Describe the sampled population(s). b. Find a point estimate for the average room rate for the Marriott hotel chain. Calculate the margin of error. c. Find a point estimate for the average room rate for the Radisson hotel chain. Calculate the margin of error. d. Find a point estimate for the average room rate for the Wyndham hotel chain. Calculate the margin of error. e. Display the results of parts \(\mathrm{b}, \mathrm{c},\) and d graphically, using the form shown in Figure \(8.5 .\) Use this display to compare the average room rates for the three hotel chains.

Short answer

In this exercise, we analyzed the nightly room rates for three hotel chains: Marriott, Radisson, and Wyndham using samples of 50 billing statements each. We calculated the point estimates and margins of error for each hotel chain at a 95% confidence level. The results show that Marriott has the highest average room rate, followed by Wyndham and Radisson. By comparing the results graphically, we can easily visualize these differences in room rates among the hotel chains.

Step-by-step solution

a. Describe the sampled population(s).

The sampled populations are the nightly room rates for three hotel chains: Marriott, Radisson, and Wyndham. For each hotel chain, we've taken a random sample of 50 billing statements which includes information about the nightly room rates.

b. Point estimate and margin of error for Marriott

For Marriott, the sample average (\(\bar{x}\)) is \(170 and the sample standard deviation (\)s\() is \)17.5. Using these data, we can calculate the point estimate and margin of error as follows: Point estimate: \(\bar{x} = \$170\) Margin of error: \(E = \frac{z \cdot s}{\sqrt{n}} = \frac{1.96 \times 17.5}{\sqrt{50}} \approx \$4.86\)

c. Point estimate and margin of error for Radisson

For Radisson, the sample average (\(\bar{x}\)) is \(145 and the sample standard deviation (\)s\() is \)10. Using these data, we can calculate the point estimate and margin of error as follows: Point estimate: \(\bar{x} = \$145\) Margin of error: \(E = \frac{z \cdot s}{\sqrt{n}} = \frac{1.96 \times 10}{\sqrt{50}} \approx \$2.77\)

d. Point estimate and margin of error for Wyndham

For Wyndham, the sample average (\(\bar{x}\)) is \(150 and the sample standard deviation (\)s\() is \)16.5. Using these data, we can calculate the point estimate and margin of error as follows: Point estimate: \(\bar{x} = \$150\) Margin of error: \(E = \frac{z \cdot s}{\sqrt{n}} = \frac{1.96 \times 16.5}{\sqrt{50}} \approx \$4.59\)

e. Compare the results graphically

To compare the results graphically, plot the point estimates for each hotel chain along with their respective margins of error (\(\bar{x} \pm E\)): 1. Marriott: \(\$ 170 \pm \$ 4.86\) (range: \(\$ 165.14\) to \(\$ 174.86\)) 2. Radisson: \(\$ 145 \pm \$ 2.77\) (range: \(\$ 142.23\) to \(\$ 147.77\)) 3. Wyndham: \(\$ 150 \pm \$ 4.59\) (range: \(\$ 145.41\) to \(\$ 154.59\)) We can observe that Marriott has the highest average room rate, followed by Wyndham and Radisson. The graphical representation allows us to visualize these differences more easily.

Key concepts

Point Estimate

When conducting a study or survey, a point estimate represents our best guess at the value of a population parameter based on our sample data. It is a single number that serves as an approximation of an unknown quantity. In the context of the exercise provided, the point estimate for the average room rate is the mean calculated from the sample of billing statements for each hotel chain.

For example, the point estimate for Marriott is based on 50 randomly selected billing statements, and it is calculated to be \(\bar{x} = \(170\). This means that \)170 is our best estimate for the average cost of a room at Marriott hotels based on the sample data. Similarly, Radisson and Wyndham chains have their point estimates calculated from their respective samples.

Margin of Error

The margin of error helps quantify the uncertainty in a point estimate. When we say there's a margin of error associated with a point estimate, we're acknowledging that the sample provides an estimate within a range rather than a precise figure. It essentially tells us how 'confident' we can be about the accuracy of the point estimate.

The formula for the margin of error is \(E = \frac{z \cdot s}{\sqrt{n}}\), where \(z\) is the z-score corresponding to the confidence level (for example, 1.96 for a 95% confidence level), \(s\) is the sample standard deviation, and \(n\) is the sample size. Increasing the sample size or having a smaller standard deviation will result in a lower margin of error. From the given exercise, the calculated margins of error for the three hotel chains—Marriott, Radisson, and Wyndham—provide a range within which we believe the true average costs are likely to fall.

Sample Standard Deviation

The sample standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In the exercise, the sample standard deviation for each hotel chain gives us an idea of how much room rates vary around the average (or point estimate). For instance, Marriott's sample standard deviation of 17.5 suggests a greater variability in room rates compared with Radisson’s 10. This can have several implications, such as a wider range of room types and prices, or more fluctuations in pricing at different times. Understanding the standard deviation in any statistical analysis is critical because it directly impacts the margin of error and, consequently, how we interpret the reliability of our point estimates.