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Chapter 11: Problem 37

A hotel chain is interested in evaluating reservation processes. Guests can reserve a room by using either a telephone system or an online system that is accessed through the hotel's web site. Independent random samples of 80 guests who reserved a room by phone and 60 guests who reserved a room online were selected. Of those who reserved by phone, 57 reported that they were satisfied with the reservation process. Of those who reserved online, 50 reported that they were satisfied. Based on these data, is it reasonable to conclude that the proportion who are satisfied is higher for those who reserve a room online? Test the appropriate hypotheses using \(\alpha=.05 .\)

Short Answer

Expert verified
Yes, it is reasonable to conclude that the proportion who are satisfied is higher for those who reserve a room online.

Step by step solution

01

State the Hypotheses

The null hypothesis \(H_0\) would be that there is no difference in satisfaction level between the two methods (p_1 = p_2), and the alternative hypothesis \(H_a\) would be that the satisfaction level is higher with the online reservation method (p_1 < p_2). Here, p_1 and p_2 are the satisfaction proportions of customers who book by phone and online, respectively.
02

Compute the Sample Proportions

First calculate the sample proportions. For phone reservations, the sample proportion (\( \hat{p}_1 \)) is 57/80 = 0.7125. For online reservations, the sample proportion (\( \hat{p}_2 \)) is 50/60 = 0.8333.
03

Computations for Z-Test

Next, calculate the pooled sample proportion and standard error. The pooled sample proportion (\( \hat{p} \)) is (57 + 50) / (80 + 60) = 107/140 = 0.7643. The standard error(SE) is calculated as \( \sqrt{\hat{p}(1-\hat{p})(1/n_1 + 1/n_2)} \) = \( \sqrt{0.7643*(1-0.7643)*(1/80 + 1/60)} \) = 0.0652.
04

Calculate Z-Statistic

The Z-statistic is calculated by subtracting the difference of the two sample proportions from zero (since under the null hypothesis we assume they are equal) and dividing by the computed standard error. Thus, Z = (0.7125 - 0.8333 - 0) / 0.0652 = -1.85.
05

Decision Rule

We reject the null hypothesis if the absolute Z value is greater than Z at the significance level (\(\alpha =0.05\)). The Z value at 0.05 significance level (1-tailed test) from the table is -1.645. Since -1.85 < -1.645, we reject the null hypothesis and conclude that a greater proportion of customers are satisfied with the online reservation process.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Z-test
The Z-test is a statistical method used to determine if there is a significant difference between the means or proportions of two groups. This test is applicable when the sample sizes are large (typically n > 30) and the distribution of the sample mean is approximately normal. In the context of this exercise, a Z-test is used to check if there is a significant difference between the satisfaction levels of guests based on their reservation method - phone or online.

The Z-test helps to **compare proportions** and find if the observed differences are due to random chance or an actual underlying cause. Here, we are comparing the proportion of satisfied customers in two independent groups: phone reservations and online reservations. By calculating the Z-statistic, which incorporates the sample proportions and the pooled standard deviation, we can infer whether there is a statistical difference between the groups.
Sample Proportion
Sample proportions represent the part of the sample that displays a certain characteristic. In hypothesis testing, sample proportions are used to estimate population proportions and make inferences about the whole group.

For the hotel example, the sample proportion is calculated for each reservation method. For phone reservations, the sample proportion is the number of satisfied customers (57) divided by the total number (80), resulting in 0.7125. For online reservations, it's 50 satisfied customers over 60, yielding 0.8333.
  • These sample proportions provide a way to compare satisfaction across groups.
  • They help determine if observed differences may indicate real differences in the populations.
By analyzing these values, we can test hypotheses about customer satisfaction.
Null Hypothesis
The null hypothesis is a fundamental concept in hypothesis testing. It proposes that there is no effect or difference in a given situation, and in this exercise, it states that the satisfaction level is the same for customers using both reservation methods.

Formally, the null hypothesis (\(H_0\)) posits that the population proportions of satisfaction (\(p_1\) for phone and \(p_2\) for online) are equal, symbolically represented as \(p_1 = p_2\).

The null hypothesis functions as a baseline statement and must be tested to be confirmed or rejected:
  • If the data suggests the null hypothesis is unlikely, it is rejected in favor of the alternative hypothesis, indicating a difference exists.
  • If the data does not provide enough evidence to reject \(H_0\), then no significant difference is deduced.
This concept ensures scientific rigor in hypothesis testing.
Pooled Sample Proportion
The pooled sample proportion is an average of the sample proportions from two independent groups. It's used in hypothesis tests when comparing two proportions, offering a single combined estimate of the sample's characteristic.

To compute the pooled proportion in our exercise, we add the satisfied customers from both samples (57 from phone reservations and 50 from online reservations) and divide by the total number of reservations (80 + 60 = 140). This gives us a pooled sample proportion of 0.7643.

The pooled sample proportion plays a critical role:
  • It helps in calculating the standard error for the Z-test.
  • It provides the basis for assessing the difference between the sample proportions and the assumed population proportion under the null hypothesis.
Thus, it is a crucial element for determining the Z-statistic and ultimately driving conclusions about the data.

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Most popular questions from this chapter

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