Question

For Exercises 7 through \(27,\) perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Airlines On-Time Arrivals The percentages of on-time arrivals for major U.S. airlines range from 68.6 to 91.1. Two regional airlines were surveyed with the following results. At \(\alpha=0.01,\) is there a difference in proportions? $$ \begin{array}{ccc}{} & {\text { Airline } \mathbf{A}} & {\text { Airline } \mathbf{B}} \\ \hline \text { No.of flights } & {300} & {250} \\ {\text { No. of on-time flights }} & {213} & {185}\end{array} $$

Short answer

Fail to reject the null hypothesis; no significant difference in proportions.

Step-by-step solution

State the Hypotheses and Identify the Claim

In hypothesis testing, we start by stating the null and alternative hypotheses. For this problem, let:- \(H_0\) (null hypothesis): There is no difference in the proportions of on-time arrivals for Airlines A and B, i.e., \(p_A = p_B\).- \(H_1\) (alternative hypothesis): There is a difference in the proportions of on-time arrivals, i.e., \(p_A eq p_B\).The claim in this case is the alternative hypothesis, \(H_1\), that there is a difference in the proportions.

Find the Critical Values

For a two-tailed test at the significance level \(\alpha = 0.01\), we will use a standard normal distribution (Z-distribution).The critical Z-values for a two-tailed test with \(\alpha = 0.01\) are approximately \(\pm 2.576\). These are the cutoff points in the Z-distribution beyond which we will reject the null hypothesis.

Compute the Test Value

To compute the test statistic for the difference in proportions, we use the formula:\[ Z = \frac{(\hat{p}_A - \hat{p}_B) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_A} + \frac{1}{n_B})}} \]First, calculate the sample proportions:- \(\hat{p}_A = \frac{213}{300} = 0.71\)- \(\hat{p}_B = \frac{185}{250} = 0.74\)The pooled proportion \(\hat{p}\) is calculated as follows:\[ \hat{p} = \frac{213 + 185}{300 + 250} = \frac{398}{550} \approx 0.7245 \]Substitute these values into the formula:\[ Z = \frac{0.71 - 0.74}{\sqrt{0.7245 (1 - 0.7245) (\frac{1}{300} + \frac{1}{250})}} \approx \frac{-0.03}{0.0401} \approx -0.748 \]

Make the Decision

Compare the calculated Z-test value with the critical values.Since \(-0.748\) does not fall outside the critical region of \(-2.576\) to \(2.576\), we fail to reject the null hypothesis. This indicates that there is insufficient evidence to support the claim that there is a difference in the proportions of on-time arrivals for Airlines A and B.

Summarize the Results

At the \(\alpha = 0.01\) significance level, we do not have enough evidence to conclude that there is a statistically significant difference in on-time arrival proportions between Airline A and Airline B.

Key concepts

Null Hypothesis

In the world of hypothesis testing, the null hypothesis, denoted as \(H_0\), is like the default assumption that there's no effect or difference in the case at hand. Imagine it as a starting point or a baseline which we assume to be true unless we find enough evidence against it.

Here's how it works in our scenario with the airlines. The null hypothesis asserts that the proportions of on-time arrivals for Airlines A and B are the same. In this context, it is represented mathematically as \(p_A = p_B\). When we say "there is no difference," we're essentially saying that any observed difference is due to random chance rather than a systematic factor.

When performing hypothesis testing:

• Initiating with \(H_0\) means you're assuming existing conditions are unaltered.
• This hypothesis requires substantial evidence to be overturned.
• Failure to reject \(H_0\) implies no significant effect detected.

Alternative Hypothesis

The alternative hypothesis, symbolized as \(H_1\) or sometimes \(H_a\), is the contrasting statement that defies the status quo proposed by the null hypothesis. It's the research hypothesis that suggests there might be a distinct effect or difference present.

For our exercise involving Airline A and B, the alternative hypothesis proclaims a difference in the proportions of on-time arrivals, mathematically expressed as \(p_A eq p_B\). This stance challenges the null hypothesis by arguing that the observed data is not likely just from random variations or noise but indicates a genuine difference.

Important points to remember:

• \(H_1\) is what the study is trying to prove.
• A statistical test aims to provide enough evidence to support \(H_1\).
• The directionality (e.g., \( >, <, eq \)) of \(H_1\) depends on the specific research question.

Critical Value

Critical values are the threshold values that define the boundaries of the region where we will reject the null hypothesis. They come from the chosen statistical distribution that represents our data, often the standard normal (Z) distribution in cases of large samples.

In our airlines scenario, since we're performing a two-tailed test with a significance level \(\alpha = 0.01\), the critical values are approximately \(\pm 2.576\). This means that our test statistic, calculated using the sample data, must exceed these values to reject \(H_0\).

Understanding critical values:

• They are influenced by the level of significance \(\alpha\), representing the probability of rejecting \(H_0\) when it's true (Type I error).
• The test's nature (one-tailed vs. two-tailed) determines the sign and number of critical values.
• Critical values serve as the cut-off points for decision-making.