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Chapter 11: Problem 9

Walking in the Airport, Part I Do people walk faster in the airport when they are departing (getting on a plane) or when they are arriving (getting off a plane)? Researcher Seth B. Young measured the walking speed of travelers in San Francisco International Airport and Cleveland Hopkins International Airport. His findings are summarized in the table. $$ \begin{array}{lcc} \text { Direction of Travel } & \text { Departure } & \text { Arrival } \\ \hline \text { Mean speed (feet per minute) } & 260 & 269 \\ \hline \begin{array}{l} \text { Standard deviation } \\ \text { (feet per minute) } \end{array} & 53 & 34 \\ \hline \text { Sample size } & 35 & 35 \\ \hline \end{array} $$ (a) Is this an observational study or a designed experiment? Why? (b) Explain why it is reasonable to use Welch's \(t\) -test. (c) Do individuals walk at different speeds depending on whether they are departing or arriving at the \(\alpha=0.05\) level of significance?

Short Answer

Expert verified
It is an observational study. Welch's t-test is applicable due to unequal variances. There is no significant difference in walking speeds (p > 0.05).

Step by step solution

01

Determine the type of study

Examine if the study involves manipulation of variables or merely observation. Describe the purpose and whether there was an intervention.
02

Title - Choose between Observational Study or Designed Experiment

In this study, Seth B. Young measured the walking speed of travelers without assigning subjects to experimental conditions. There was no manipulation of variables; therefore, this is an observational study.
03

Understanding Welch's t-test applicability

Analyze the appropriateness of using Welch's t-test given the dataset's characteristics. Assess the assumptions and conditions needed for Welch's t-test.
04

Title - Justify Use of Welch's t-test

Welch's t-test is used for comparing the means of two samples that may have different variances and sample sizes. Here, the sample sizes for Departure and Arrival are equal, but the standard deviations are different (53 and 34, respectively). Thus, it is reasonable to use Welch's t-test to account for the unequal variances.
05

Hypothesis testing at 伪 = 0.05

Formulate the null and alternative hypotheses. Calculate the test statistic and compare it with the critical value or use the p-value to make a decision.
06

Title - Perform Hypothesis Testing

We want to test if the means are different:Null Hypothesis: \ H_0: \ \text{碌}_{\text{Departure}} = \text{碌}_{\text{Arrival}} Alternative Hypothesis: \ H_1: \ \text{碌}_{\text{Departure}} 鈮� \text{碌}_{\text{Arrival}}The formula for the test statistic (t) in Welch's t-test is given by:\[ t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]where:\( \bar{X}_1 \) = 260, \( \bar{X}_2 \) = 269,\( s_1 \) = 53, \( s_2 \) = 34,\( n_1 \) = 35, \( n_2 \) = 35.\[ t = \frac{260 - 269}{\sqrt{\frac{53^2}{35} + \frac{34^2}{35}}} \approx -0.885 \]Now compare the computed t-value with the critical t-value from the t-distribution table at \(\alpha=0.05\). With degrees of freedom approximated by:\[ df \approx \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}} \approx 55.42 \]The critical value for t at \(\alpha=0.05\) two-tailed test with 55 degrees of freedom is approximately 2.004.Since \(|-0.885| \) < 2.004, we fail to reject the null hypothesis. Thus, there is no significant evidence at \(\alpha=0.05\) to indicate a difference in walking speeds between departing and arriving travelers.

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Key Concepts

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

observational study
An observational study focuses on observing and recording data without manipulating any variables. In this study, Seth B. Young measured walking speeds of travelers at two airports, simply recording the data without altering travelers' conditions or behaviors. Observational studies are often used when direct manipulation would be unethical or impractical. By not influencing participants, observational studies can provide naturalistic insights, though they may also be subject to confounding variables that can affect results.
Welch's t-test
Welch's t-test is a statistical test used to compare the means of two samples, particularly when they have unequal variances and/or sample sizes. It is a more robust alternative to the traditional Student's t-test.
In this exercise, the sample sizes for departing and arriving travelers are the same (both 35), but the standard deviations differ (53 and 34). This difference in variances makes Welch's t-test appropriate since it accounts for the unequal variability in the two groups.
The main equation for Welch's t-test is:
\[ t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]
where:
  • \( \bar{X}_1 \) = mean of the first sample (260 feet per minute for Departure)
  • \( \bar{X}_2 \) = mean of the second sample (269 feet per minute for Arrival)
  • \( s_1 \) = standard deviation of the first sample (53)
  • \( s_2 \) = standard deviation of the second sample (34)
  • \( n_1 \) and \( n_2 \) are sample sizes (both 35)
hypothesis testing
Hypothesis testing is a statistical method used to make decisions or inferences about a population based on sample data. It involves comparing a null hypothesis (\( H_0 \)) with an alternative hypothesis (\( H_1 \)) to see if there is significant evidence to reject \( H_0 \).
In this exercise, we test whether travelers' walking speeds are different when they are departing versus arriving:
Null Hypothesis: \( H_0: \text{碌}_{Departure} = \text{碌}_{Arrival} \)
Alternative Hypothesis:\( H_1: \text{碌}_{Departure} eq \text{碌}_{Arrival} \)
The test statistic, using Welch's t-test, is calculated as:
\[ t = \frac{260 - 269}{\sqrt{\frac{53^2}{35} + \frac{34^2}{35}}} \approx -0.885 \]
Next, we compare the calculated t-value with a critical value from the t-distribution table at the significance level \( \alpha = 0.05 \).
With about 55 degrees of freedom (df), the critical t-value for a two-tailed test is approximately 2.004. Since \( |-0.885| < 2.004 \), we fail to reject the null hypothesis. This means there is no significant evidence at the \( \alpha = 0.05 \) level to indicate that walking speeds differ between departing and arriving travelers.

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Most popular questions from this chapter

Elapsed Time to Earn a Bachelor's Degree A researcher with the Department of Education followed a cohort of students who graduated from high school in \(1992,\) monitoring the progress the students made toward completing a bachelor's degree. One aspect of his research was to determine whether students who first attended community college took longer to attain a bachelor's degree than those who immediately attended and remained at a 4-year institution. The data in the table summarize the results of his study. $$ \begin{array}{lcc} & \begin{array}{l} \text { Community College } \\ \text { to Four-Year Transfer } \end{array} & \text { No Transfer } \\ \hline n & 268 & 1145 \\ \hline \begin{array}{l} \text { Sample mean time to } \\ \text { graduate, in years } \end{array} & 4.43 \\ \hline \begin{array}{l} \text { Sample standard } \\ \text { deviation time to } \\ \text { graduate, in years } \end{array} & 1.162 & 1.015 \\ \hline \end{array} $$ (a) What is the response variable in this study? What is the explanatory variable? (b) Explain why this study can be analyzed using the methods of this section. (c) Does the evidence suggest that community college transfer students take longer to attain a bachelor's degree? Use an \(\alpha=0.01\) level of significance. (d) Construct a \(95 \%\) confidence interval for \(\mu\) community college \(\mu_{\text {no transfer }}\) to approximate the mean additional time it takes to complete a bachelor's degree if you begin in community college. (e) Do the results of parts (c) and (d) imply that community college causes you to take extra time to earn a bachelor's degree? Cite some reasons that you think might contribute to the extra time to graduate.

John has an online company that sells custom rims for cars and designed two different web pages that he wants to use to sell his rims online. However, he cannot decide which page to go with, so he decides to collect data to see which site results in a higher proportion of sales. He hires a firm that has the ability to randomly assign one of his two web page designs to potential customers. With web page design I, John secures a sale from 54 out of 523 hits to the page. With web page design II, John secures a sale from 62 out of 512 hits to the page. (a) What is the response variable in this study? What is the explanatory variable? (b) Based on these results, which web page, if any, should John go with? Why? Note: This problem is based on the type of research done by Adobe Test \& Target.

Aluminum Bottles The aluminum bottle, first introduced in 1991 by CCL Container for mainly personal and household items such as lotions, has become popular with beverage manufacturers. Besides being lightweight and requiring less packaging, the aluminum bottle is reported to cool faster and stay cold longer than typical glass bottles. A small brewery tests this claim and obtains the following information regarding the time (in minutes) required to chill a bottle of beer from room temperature \(\left(75^{\circ} \mathrm{F}\right)\) to serving temperature \(\left(45^{\circ} \mathrm{F}\right) .\) Construct and interpret a \(90 \%\) confidence interval for the mean difference in cooling time for clear glass versus aluminum. $$ \begin{array}{lcc} & \text { Clear Glass } & \text { Aluminum } \\ \hline \text { Sample size } & 42 & 35 \\ \hline \begin{array}{l} \text { Mean time to } \\ \text { chill (minutes) } \end{array} & 133.8 & 92.4 \\ \hline \begin{array}{l} \text { Sample standard } \\ \text { deviation (minutes) } \end{array} & 9.9 & 7.3 \\ \hline \end{array} $$

In Problems 3鈥�8, determine whether the sampling is dependent or independent. Indicate whether the response variable is qualitative or quantitative. A sociologist wishes to compare the annual salaries of married couples in which both spouses work and determines each spouse鈥檚 annual salary.

Perform the appropriate hypothesis test. A random sample of size \(n=13\) obtained from a population that is normally distributed results in a sample mean of 45.3 and sample standard deviation of \(12.4 .\) An independent sample of size \(n=18\) obtained from a population that is normally distributed results in a sample mean of 52.1 and sample standard deviation of 14.7 . Does this constitute sufficient evidence to conclude that the population means differ at the \(\alpha=0.05\) level of significance?
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