Question

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

To decide whether or not it is reasonable to conclude that the proportion who are satisfied is higher for those who reserve a room online, compare the p-value to the significance level. If the p-value < α, the evidence supports the claim that the proportion of customers who are satisfied is higher for online reservations.

Step-by-step solution

Identify a sample proportion for each group

The sample proportion \(p_1\) for phone reservations is 57 out of 80, which equals 0.7125. The sample proportion \(p_2\) for online reservations is 50 out of 60, which equals 0.8333.

Formulate the Null and Alternative Hypotheses

The null hypothesis \(H_0: p_1 = p_2\), implies that the proportion of guests satisfied with the reservation system is the same for both phone and online systems. The alternative hypothesis \(H_a: p_1 < p_2\), implies that the proportion of guests satisfied with the online reservation system is greater than that of the phone reservation system.

Compute the pooled sample proportion and Standard Error

The pooled sample proportion \( \hat{p}\) is calculated by adding the number of 'successes' (satisfied customers) in both groups and dividing by the total number of observations in both groups. The standard error (SE) of the difference in sample proportions is calculated using the formula \(SE = \sqrt{ \hat{p} ( 1 - \hat{p} ) (1/n_1 + 1/n_2)}\), where \(n_1\) and \(n_2\) are the sample sizes for each group.

Calculate the z-score

The z-score is calculated by subtracting the null hypothesis value (which is 0 in this case as \(H_0\) states that the population proportions are equal) from the difference in sample proportions, and then dividing that difference by the standard error.

Find the p-value

The p-value is the probability of obtaining a z-score equal to or more extreme than the observed z-score, assuming the null hypothesis is true. The p-value is found using a standard normal z distribution table.

Make a decision

If the p-value is less than the significance level (α=0.05), reject the null hypothesis.

Key concepts

Sample Proportion

A sample proportion is a statistic that gives us an estimate of a population parameter. It is calculated by dividing the number of successful outcomes by the total number in the sample. For example, in the hotel reservation study, sample proportion helps us understand the percentage of satisfied customers.

• For phone reservations: Out of 80 guests, 57 were satisfied. So, the sample proportion is calculated as: \[ p_1 = \frac{57}{80} = 0.7125 \]
• For online reservations: Out of 60 guests, 50 were satisfied. So, the sample proportion is:\[ p_2 = \frac{50}{60} = 0.8333 \]

These proportions give us a sense of how many people on average are satisfied with each reservation method.

Null and Alternative Hypotheses

In hypothesis testing, we start by stating two opposing statements: the null hypothesis (denoted as \(H_0\)) and the alternative hypothesis (denoted as \(H_a\)). This initial step frames our investigation and helps us understand what we are testing.For the reservation process:

• Null Hypothesis \(H_0: p_1 = p_2\): This proposes that there is no difference in satisfaction between phone and online reservations.
• Alternative Hypothesis \(H_a: p_1 < p_2\): Suggests that a higher proportion are satisfied with online reservations compared to phone reservations.

These hypotheses allow us to test if the observed differences in sample proportions are statistically significant.

Pooled Sample Proportion

The pooled sample proportion combines data from both groups to provide an overall estimate of success across the samples. Calculating the pooled sample proportion is useful when testing for differences between two proportions.In our hotel reservations case:We calculate the pooled sample proportion \(\hat{p}\) by taking the total number of satisfied customers from both groups and dividing it by the total number of observations from both groups.\[ \hat{p} = \frac{57 + 50}{80 + 60} = \frac{107}{140} = 0.7643 \]This combined proportion helps standardize our results and is used to calculate the standard error needed to test our hypotheses.

p-value

The p-value plays a crucial role in deciding whether to reject the null hypothesis. It represents the probability of observing a test statistic as extreme as or more extreme than the one observed under the assumption that the null hypothesis is true.In this study:

• We compute the test statistic, usually a z-score, based on our sample data.
• The p-value is found using the standard normal distribution, showing the likelihood of the observed difference if \(H_0\) is true.

If the p-value is less than our significance level (in this case, \(\alpha = 0.05\)), then we reject the null hypothesis. This tells us that there is statistically significant evidence to suggest a difference in satisfaction between phone and online reservations.