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 greater for those who reserve a room online? Test the appropriate hypotheses using a significance level of \(0.05 .\)

Short answer

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

Step-by-step solution

Define the variables

Let's denote the proportion of people who are satisfied with online reservation as \(P_1\) and those by phone as \(P_2\). Let \(P = P_1 - P_2\). We need to test if \(P\) is greater than 0.

Formulate Null and Alternate Hypotheses

Null Hypothesis (\(H_0\)): There is no difference between the proportions. Therefore, \(P = 0\). Alternate Hypothesis (\(H_1\)): The proportion of people satisfied with online reservation is greater than those by phone, so \(P > 0\).

Compute proportions and standard error

First, calculate the sample proportions. For online (\(P_1\)): 50 out of 60 are satisfied, so \(P_1 = 50/60 = 0.8333\). For phone (\(P_2\)): 57 out of 80 are satisfied, \(P_2 = 57/80 = 0.7125\). Calculate the overall proportion \(P_{combined}\), which equals to the total guests satisfied divided by the total guests. So \(P_{combined} = (50+57) / (60+80) = 0.7679\). Now, calculate standard error \(SE\) using the formula: \(SE = \sqrt{P_{combined}*(1-P_{combined})*(1/n_1 + 1/n_2)}\), where \(n_1\) and \(n_2\) are the number of guests online and by phone respectively. This gives \(SE = \sqrt{0.7679*(1-0.7679)*(1/60 + 1/80)} = 0.0633\).

Calculate Test Statistic

The test statistic \(Z\) should be calculated using the formula: \(Z = (P-P_{H_0})/SE\), where \(P_{H_0}\) is the proposed difference under the null hypothesis which is 0. This yields \(Z = (0.8333 - 0.7125)/0.0633 = 1.9058\).

Reject or Fail to Reject the Null Hypothesis

The given significance level is 0.05. Find the Z value corresponding to this significance level from the Z-table or normal distribution table. Since it's a one-tailed test, look for Z value at 0.95 (1 - 0.05) in the Z-table. It's approximately 1.645. The calculated Z value (1.9058) is more than this value, so we reject the null hypothesis. There is enough evidence to conclude that the proportion of people who are satisfied is greater for those who reserve a room online.

Key concepts

Proportion Test

A proportion test is a statistical tool used to evaluate whether there is a significant difference between the proportions of a certain characteristic within two groups. In the context of our exercise, this test helps us determine if the proportion of hotel guests satisfied with their reservation process differs between two reservation methods, namely online versus by phone.

When conducting a proportion test, we compare our sample data to the null hypothesis, which in this case suggests that there is no divergence in satisfaction between the two reservation methods. If the data significantly deviates from what the null hypothesis expects, we may have grounds to accept the alternate hypothesis – which postulates the existence of a difference in satisfaction levels. Calculating the Z value in relation to the standard error pinpoints where our sample proportion difference lies on the normal distribution, guiding our decision to reject or not reject the null hypothesis.

Null Hypothesis

The null hypothesis, symbolized as \(H_0\), is a fundamental component of hypothesis testing, serving as a statement that there is no effect or no difference to be observed. It acts as a baseline or a skeptical perspective, positing that any observed variation in the data is due to chance rather than a systematic effect.

In our reservation satisfaction exercise, the null hypothesis asserts that the satisfaction rate for online and phone reservations is the same (i.e., there is no difference). This forms a benchmark against which we measure the observed sample proportions. If our test results suggest that the observed data is highly unlikely under the null hypothesis, we may reject \(H_0\) in favor of the alternate hypothesis. Importance lies in the ability to quantify the evidence against \(H_0\) to avoid mistakenly rejecting it when it is actually true (type I error).

Alternate Hypothesis

The alternate hypothesis, denoted by \(H_1\) or \(H_a\), is the counter-claim to the null hypothesis in the testing framework. It represents a researcher's belief that there is a genuine effect or difference to be detected. In the given scenario, the alternate hypothesis posits that the proportion of guests satisfied with online reservations is greater than those satisfied with phone reservations.

Establishing the alternate hypothesis is crucial for directing the research and understanding what kind of evidence is being sought. If the null hypothesis is rejected, it implies support for the alternate hypothesis within the confidence level set by the significance level. However, it's important not to confuse rejection of the null with irrefutable proof of the alternate hypothesis; we're just indicating that the data supports the alternate hypothesis over the null given the evidence at hand.

Standard Error

Standard error (SE) is a measure that indicates the variability of the sampling distribution of a statistic, like a mean or proportion. It allows researchers to judge how much the sample statistic might fluctuate if the experiment were repeated multiple times.

In hypothesis testing for proportions, SE plays a critical role as it is used in calculating the test statistic (like the Z-score). The standard error is influenced by both the sample size and the variability of the data. In large samples, we expect the standard error to be smaller, meaning the statistics are likely to be closer to the population parameter. The exercise shows how to calculate the SE for the difference in proportions, relying on the combined proportion and the size of both groups. As the SE decreases, any observed differences in the sample become more statistically significant.

Z-Test

A Z-test is a statistical method that determines if there is a significant difference between sample and population parameters or between the parameters of two samples. The test is based on the assumption that the data points are normally distributed and that the population variance is known or estimated sufficiently by the sample.

In the context of our exercise, the Z-test evaluates whether the observed difference in satisfaction proportions is significant. This requires calculating a Z score, which measures how many standard errors our sample proportion difference is from the null hypothesis' proportion difference. If the Z score exceeds a critical value on the normal distribution curve — determined by the significance level — we reject the null hypothesis. The exercise illustrates a one-tailed Z-test because the alternative hypothesis specifies a direction of the difference (greater satisfaction in online reservations).