Question

Perform the following steps. Assume that all variables are normally distributed. a. State the hypotheses and identify the claim. b. Find the critical value. c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Cities were randomly selected from the list of the 50 largest cities in the United States (based on population). The areas of each in square miles are shown. Is there sufficient evidence to conclude that the variance in area is greater for eastern cities than for western cities at \(\alpha=0.05 ?\) At \(\alpha=0.01 ?\) $$ \begin{array}{lc|ll} &{\text { Eastern }} & {\text { Western }} \\ \hline \text { Atlanta, GA } & 132 & \text { Albuquerque, NM } & 181 \\ \text { Columbus, OH } & 210 & \text { Denver, CO } & 155 \\ \text { Louisville, KY } & 385 & \text { Fresno, CA } & 104 \\ \text { New York, NY } & 303 & \text { Las Vegas, NV } & 113 \\ \text { Philadelphia, PA } & 135 & \text { Portland, OR } & 134 \\ \text { Washington, DC } & 61 & \text { Seattle, WA } & 84 \\ \text { Charlotte, NC } & 242 & & \end{array} $$

Short answer

At \(\alpha=0.05\), we have enough evidence to support the claim, whereas at \(\alpha=0.01\), we do not.

Step-by-step solution

State the Hypotheses

In this step, we identify the null and alternative hypotheses. Let \( \sigma^2_E \) be the variance in area for eastern cities, and \( \sigma^2_W \) for western cities. We want to test if \( \sigma^2_E > \sigma^2_W \).- Null hypothesis \(H_0\): \( \sigma^2_E \leq \sigma^2_W \)- Alternative hypothesis \(H_a\): \( \sigma^2_E > \sigma^2_W \)The claim we are testing is the alternative hypothesis, which states that the variance in area is greater for eastern cities.

Find the Critical Value

To find the critical value, we use the F-distribution because we are comparing variances. The critical value depends on the significance level \(\alpha\) and the degrees of freedom for each group.- Eastern cities: 7 observations (\(n_1 = 7\)), so \(df_1 = n_1 - 1 = 6\).- Western cities: 6 observations (\(n_2 = 6\)), so \(df_2 = n_2 - 1 = 5\).Using \(\alpha = 0.05\) and \(\alpha = 0.01\), we find the critical value from the F-distribution tables:- For \(\alpha = 0.05\), \(F_{critical} = F(6, 5, 0.05)\)- For \(\alpha = 0.01\), \(F_{critical} = F(6, 5, 0.01)\)Using standard F-tables, the critical value at \(\alpha = 0.05\) is approximately 4.95, and at \(\alpha = 0.01\) is approximately 10.97.

Compute the Test Value

To compute the test statistic, we calculate the sample variances and then form the F-ratio.- First, calculate the sample variances: - Eastern variance \(s^2_E = \frac{\sum (x_i - \bar{x})^2}{n_1 - 1}\) - Western variance \(s^2_W = \frac{\sum (x_i - \bar{x})^2}{n_2 - 1}\)- After numerical calculations, the variances are approximately: - \(s^2_E \approx 11312.17\) - \(s^2_W \approx 1422.17\)- Compute the F-ratio: \[ F = \frac{s^2_E}{s^2_W} = \frac{11312.17}{1422.17} \approx 7.95 \]

Make the Decision

Compare the calculated F-statistic with the critical values:- At \(\alpha=0.05\): - Calculated \(F = 7.95\) exceeds the critical value of 4.95. - Therefore, reject \(H_0\).- At \(\alpha=0.01\): - Calculated \(F = 7.95\) does not exceed the critical value of 10.97. - Do not reject \(H_0\).For \(\alpha = 0.05\), there is evidence to support the claim, but not for \(\alpha = 0.01\).

Summarize the Results

In conclusion, there is sufficient evidence at the \(\alpha=0.05\) level to suggest that the variance in area is greater for eastern cities than for western cities. However, at the \(\alpha=0.01\) level, there is not enough evidence to make this claim.

Key concepts

F-distribution

When comparing variances in hypothesis testing, the F-distribution plays a crucial role. The F-distribution is a continuous probability distribution that arises in the context of variance ratio tests. It is asymmetric and always positive, as variances cannot be negative. This distribution is characterized by two parameters: degrees of freedom for the numerator and denominator.

In our context, we use the F-distribution to compare the variance of eastern cities to the variance of western cities. The degrees of freedom for the numerator (\(df_1 \)) are determined by the sample size of the eastern cities minus one, and the degrees of freedom for the denominator (\(df_2 \)) by the sample size of the western cities minus one.

Using the F-distribution table, we can find critical values which help in deciding whether the observed variance difference is statistically significant.

Variance Comparison

Variance comparison is key in determining how much variation exists within datasets. It shows how much the data points in each group differ from their respective means.

In hypothesis testing, especially with variances, we formulate hypotheses to test if one variance is significantly greater than another. For instance, comparing the variance of city areas in the eastern versus western United States, we form null and alternative hypotheses:

• Null hypothesis (\(H_0\)): No greater variance in eastern cities compared to western cities (\(\sigma^2_E \leq \sigma^2_W\)).
• Alternative hypothesis (\(H_a\)): Greater variance in eastern cities (\(\sigma^2_E > \sigma^2_W\)).

This comparison helps to identify if there is a significant difference based on statistical calculations.

Critical Value

Critical values are certain points on the F-distribution curve that define the threshold for deciding whether to reject the null hypothesis. They depend on the chosen significance level and the degrees of freedom for both groups being compared.

Using significance levels such as \(\alpha = 0.05\) and \(\alpha = 0.01\), we locate critical values using F-distribution tables or statistical software for given degrees of freedom.

For this particular case, the calculated critical values are approximately 4.95 and 10.97 for \(\alpha = 0.05\) and \(\alpha = 0.01\), respectively. If our computed F-test statistic exceeds the critical value, we reject the null hypothesis under the specified significance level.

Test Statistic

The test statistic in this scenario is derived from the F-ratio, which is a ratio of two variances. It is formed by dividing one sample's variance by another, specifically the variance of eastern cities by the variance of western cities.

To find the test statistic, first, compute each sample variance. Using the variances, the F-ratio is: \[ F = \frac{s^2_E}{s^2_W} \]

For our data, numerically calculated sample variances provide an F-statistic of approximately 7.95. This statistic is essential as it is used along with critical values to make a decision regarding the null hypothesis.

Significance Level

Significance levels, denoted by \(\alpha\), indicate the probability of rejecting the null hypothesis when it is actually true. These levels are crucial as they define the "risk" we are willing to take in hypothesis testing.

Commonly used significance levels are \(\alpha = 0.05\) and \(\alpha = 0.01\), showing a 5% or 1% risk, respectively. A lower significance level means we require stronger evidence to reject the null hypothesis.

In our test, at \(\alpha = 0.05\), we have enough evidence to support the claim of greater variance for eastern cities, but at \(\alpha = 0.01\), the claim lacks sufficient evidence. Adjusting significance levels can thus, impact the interpretation of the results.