Question

Stopping Distances of Automobiles A researcher wishes to see if the stopping distance for midsize automobiles is different from the stopping distance for compact automobiles at a speed of 70 miles per hour. The data are shown for two random samples. At \(\alpha=0.10,\) test the claim that the stopping distances are the same. If one of your safety concerns is stopping distance, will it make a difference which type of automobile you purchase? $$\begin{array}{l|cccccccccc}\text { Automobile } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline \text { Midsize } & 188 & 190 & 195 & 192 & 186 & 194 & 188 & 187 & 214 & 203 \\\\\hline \text { Compact } & 200 & 211 & 206 & 297 & 198 & 204 & 218 & 212 & 196 & 193\end{array}$$

Short answer

There is a significant difference in stopping distances at \(\alpha = 0.10\); hence, your safety concern should consider the type of automobile.

Step-by-step solution

State the Hypotheses

We begin by stating the null hypothesis and the alternative hypothesis for our two-sample t-test. The null hypothesis \(H_0\) states that there is no difference in the average stopping distances between midsize and compact automobiles. The alternative hypothesis \(H_a\) states that there is a difference. Therefore, \(H_0: \mu_1 = \mu_2\) and \(H_a: \mu_1 eq \mu_2\), where \(\mu_1\) and \(\mu_2\) represent the mean stopping distances for midsize and compact automobiles, respectively.

Determine the Significance Level

The problem states a significance level \(\alpha = 0.10\). This determines the threshold at which we will reject the null hypothesis.

Collect Sample Data and Calculate Means

Calculate the sample means for both the midsize and compact automobiles. Midsize sample mean \( \bar{x}_1 \) is calculated by averaging the given data points: 188, 190, 195, 192, 186, 194, 188, 187, 214, 203. Compact sample mean \( \bar{x}_2 \) is calculated from: 200, 211, 206, 197, 198, 204, 218, 212, 196, 193.

Calculate Sample Standard Deviations

Find the standard deviations for both samples. Use the formula \( s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2} \) where \(n\) is the number of observations in each sample and \(\bar{x}\) is the sample mean. Perform this calculation for both the midsize and compact samples.

Perform a Two-Sample T-test

With both sample means and standard deviations, compute the t-statistic using the formula: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \] where \(s_1\) and \(s_2\) are the standard deviations, \(n_1\) and \(n_2\) are the sample sizes.

Calculate Degrees of Freedom and Critical Value

Use the degrees of freedom formula for a two-sample t-test: \[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}} \]. Then, find the critical t-value from the t-distribution table at \(\alpha = 0.10\) and the calculated degrees of freedom.

Make a Decision

Compare the calculated t-statistic with the critical value. If the absolute value of the t-statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Conclusion

Depending on the decision from Step 7, summarize the findings. If we rejected the null hypothesis, conclude that the stopping distances differ between midsize and compact automobiles. If not, conclude that there is no significant evidence to suggest a difference.

Key concepts

Hypothesis Testing

Hypothesis testing is a fundamental aspect of statistics that helps us decide between two competing claims, called hypotheses, based on our data. In the context of the two-sample t-test, we start by defining two hypotheses: the null hypothesis and the alternative hypothesis.

The **null hypothesis** (denoted as \(H_0\)) posits that there is no effect or difference. For our exercise, this means we assume the stopping distances of midsize and compact automobiles are the same. Mathematically, this is expressed as \(H_0: \mu_1 = \mu_2\), where \(\mu_1\) and \(\mu_2\) represent the true mean stopping distances for midsize and compact cars, respectively.

Conversely, the **alternative hypothesis** (denoted as \(H_a\)) suggests that there is a difference. So, \(H_a: \mu_1 eq \mu_2\) indicates that one of the automobiles has a different average stopping distance. By comparing these hypotheses, we use our data to determine whether we have enough evidence to reject the null hypothesis in favor of the alternative.

Significance Level

The **significance level**, denoted by \(\alpha\), is a threshold that helps us evaluate our hypothesis testing results. It provides a yardstick for how extreme our test results must be before we decide to reject the null hypothesis.

In our problem, the significance level is set at \(\alpha = 0.10\). This implies a 10% risk level when it comes to a Type I error, which occurs if we mistakenly reject a true null hypothesis.

To put it simply, if the computed likelihood (or p-value) of observing such data by chance under the null hypothesis is less than our 10% threshold, we will reject the null hypothesis. This is a balance between confidence and caution, and in our exercise, suggests moderate openness to finding a significant difference.

Sample Standard Deviation

The **sample standard deviation** is a measure of how much individual data points differ from the sample mean. It provides insight into the variability within our samples.

To compute the sample standard deviation, we use the formula:

• \( s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2} \)

where:

• \(n\) is the number of observations in the sample.
• \(x_i\) represents each data point.
• \(\bar{x}\) is the sample mean.

For our exercise, we calculate this for each group, midsize and compact, to understand the spread of stopping distances and to use it further in calculating the t-statistic. This information is crucial for understanding whether observed differences are meaningful or simply due to random variation.

T-statistic Calculation

The **t-statistic** is a crucial value in hypothesis testing. It quantifies the difference between your sample results, scaled by the variability present within your samples. Essentially, it tells you how far your sample mean is from the proposed population mean, in standard deviation units.

To calculate the t-statistic for a two-sample test, we use the formula:

• \( t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \)

where:

• \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means.
• \(s_1\) and \(s_2\) are the sample standard deviations.
• \(n_1\) and \(n_2\) are the respective sample sizes.

This statistic helps us determine if the observed differences in means are statistically significant or occur by random chance. In simple terms, a larger absolute t-value suggests a larger difference between the groups.

Degrees of Freedom

The **degrees of freedom** (df) are an essential concept in statistics, particularly when conducting hypothesis tests involving t-tests. Degrees of freedom refer to the number of independent values that can vary in your statistical calculation without restriction. They are crucial for determining the appropriate distribution to use when assessing the significance of your t-test results.

For our two-sample t-test, the degrees of freedom are calculated using the formula:\[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}} \].

This equation considers the sizes and variances of the two samples, providing a df value suited for the unequal variance scenario often encountered in real-world data. Finding the correct df allows us to reference the right t-distribution when determining the critical t-value to compare with our calculated t-statistic, ultimately helping facilitate our decision in hypothesis testing.