91Ó°ÊÓ

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9 - 1 along with "Table" answers based on Table \(A\) - 3 with df equal to the smaller of \(\boldsymbol{n}_{I}-\boldsymbol{I}\) and \(\boldsymbol{n}_{2}-\boldsymbol{I} .\) ) Car and taxi Ages When the author visited Dublin, Ireland (home of Guinness Brewery employee William Gosset, who first developed the \(t\) distribution), he recorded the ages of randomly selected passenger cars and randomly selected taxis. The ages can be found from the license plates. (There is no end to the fun of traveling with the author.) The ages (in years) are listed below. We might expect that taxis would be newer, so test the claim that the mean age of cars is greater than the mean age of taxis. $$\begin{array}{l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline \text { Car Ages } & 4 & 0 & 8 & 11 & 14 & 3 & 4 & 4 & 3 & 5 & 8 & 3 & 3 & 7 & 4 & 6 & 6 & 1 & 8 & 2 & 15 & 11 & 4 & 1 & 6 & 1 & 8 \\ \hline \text { Taxi Ages } & 8 & 8 & 0 & 3 & 8 & 4 & 3 & 3 & 6 & 11 & 7 & 7 & 6 & 9 & 5 & 10 & 8 & 4 & 3 & 4 & & & & & & \\ \hline\end{array}$$

Step by step solution

01

- Calculate Sample Statistics

Calculate the sample mean and standard deviation for both car ages and taxi ages. Let's denote the sample means as \(\bar{X}_{\text{cars}}\) and \(\bar{X}_{\text{taxis}}\), and the sample standard deviations as \(\text{SD}_{\text{cars}}\) and \(\text{SD}_{\text{taxis}}\).

02

- State the Hypotheses

Formulate the null hypothesis \(H_0\): \(\bar{X}_{\text{cars}} \leq \bar{X}_{\text{taxis}}\) and the alternative hypothesis \(H_1\): \(\bar{X}_{\text{cars}} > \bar{X}_{\text{taxis}}\).

03

- Determine the Test Statistic

Use the formula for the test statistic for two independent samples: $$t = \frac{\bar{X}_{1} - \bar{X}_{2}}{\sqrt{\frac{SD_{1}^{2}}{n_{1}} + \frac{SD_{2}^{2}}{n_{2}}}}$$ where \(n_{1}\) and \(n_{2}\) are the sample sizes of cars and taxis, respectively.

04

- Calculate Degrees of Freedom

According to the information, use the smaller of \(n_{\text{cars}}-1\) and \(n_{\text{taxis}}-1\) as the degrees of freedom \(df\).

05

- Determine the Critical Value

Using the degrees of freedom and the significance level (typically 0.05, but can vary depending on the context), find the critical value from the t-distribution table.

06

- Compare Test Statistic to Critical Value

Compare the calculated test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis.

07

- Make a Decision

Depending on the comparison in Step 6, conclude whether to reject the null hypothesis and accept the alternative hypothesis that the mean age of cars is greater than that of taxis.

08

- Interpret the Results

Based on the decision made in Step 7, interpret the results in the context of the problem. If \(H_0\) is rejected, we have sufficient evidence to support the claim that the mean age of cars is greater than the mean age of taxis.

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Key Concepts

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

Independent Samples

An independent sample involves observing or measuring two groups independently. In our exercise, the ages of cars and taxis represent two separate, unconnected groups. Each group's sample should be randomly chosen to ensure the results' validity.

• The car and taxi ages data were collected independently.
• No car or taxi's age influenced the data of the other group.

This independence is key when performing calculations, as assumptions about the samples only hold if they are truly independent.

Normally Distributed Populations

A normally distributed population follows a bell curve, with most data points clustering around the mean.

Our exercise assumed that car and taxi ages come from normally distributed populations. This assumption is crucial for the validity of the t-test:

• It allows the use of specific statistical properties essential for calculating the t-statistic.
• Without this normal distribution, the test results may not be reliable.

For small sample sizes, this normality assumption becomes even more important.

Degrees of Freedom

Degrees of freedom (df) are a way to describe the number of values in a calculation that are free to vary. In our two-sample t-test, df helps determine crucial test values:

• The formula for df is given by the smaller of \( n_{\text{cars}} - 1 \) or \( n_{\text{taxis}} - 1 \).
• For instance, if \( n_{\text{cars}} = 27 \) and \( n_{\text{taxis}} = 20 \), the df would be calculated as the lesser between \( 27 - 1 = 26 \) and \( 20 - 1 = 19 \), which is 19.

Properly determining df ensures that the critical values from the t-distribution are accurate, leading to valid conclusions.

Null Hypothesis

The null hypothesis (\( H_0 \)) is a default statement that there is no effect or no difference. In our exercise:

• The null hypothesis is that the mean age of cars is less than or equal to the mean age of taxis: \( H_0 : \bar{X}_{\text{cars}} \leq \bar{X}_{\text{taxis}} \).

We test this hypothesis against the alternative hypothesis \( H_1 \) (that cars are older than taxis). Rejecting \( H_0 \) would support the claim that cars are generally older than taxis.

Critical Value

A critical value in hypothesis testing determines the cutoff point to decide if the null hypothesis should be rejected. For our case:

• Use the t-distribution table.
• Match the degrees of freedom (df).
• Identify the value corresponding to the chosen significance level (e.g., 0.05).

Compare the calculated test statistic from the data to this critical value:

• If the test statistic is greater than the critical value, reject \( H_0 \).
• If it's lower, do not reject \( H_0 \).

This approach ensures we make informed decisions based on statistical evidence.