Question

An article in American Demographics investigated consumer habits at the mall. We tend to spend the most money shopping on the weekends, and, in particular, on Sundays from 4 to 6 P.M. Wednesday morning shoppers spend the least! \({ }^{19}\) Suppose that a random sample of 20 weekend shoppers and a random sample of 20 weekday shoppers were selected, and the amount spent per trip to the mall was recorded. $$ \begin{array}{lcc} & \text { Weekends } & \text { Weekdays } \\ \hline \text { Sample Size } & 20 & 20 \\ \text { Sample Mean (\$) } & 78 & 67 \\ \text { Sample Standard Deviation (\$) } & 22 & 20 \end{array} $$ a. Is it reasonable to assume that the two population variances are equal? Use the \(F\) -test to test this hypothesis with \(\alpha=.05\). b. Based on the results of part a, use the appropriate test to determine whether there is a difference in the average amount spent per trip on weekends versus weekdays. Use \(\alpha=.05\).

Short answer

Answer: No, there is no significant difference between the average amount spent per trip on weekends versus weekdays at the α=0.05 level of significance.

Step-by-step solution

Part a: Testing equality of variances using F-test

Formulate the hypothesis: - Null hypothesis (H0): Variances of two populations are equal, \(σ_1^2 = σ_2^2\). - Alternative hypothesis (H1): Variances of two populations are not equal, \(σ_1^2 ≠ σ_2^2\). For the F-test, we will use the given samples and the significance level α=0.05: 1. Calculate the F statistic. F = (Sample variance 1) / (Sample variance 2) = \((\frac{22^2}{20^2})\) = \(\frac{484}{400} = 1.21\). 2. Determine the degrees of freedom for both samples. df1 = n1 - 1 = 20 - 1 = 19 and df2 = n2 - 1 = 20 - 1 = 19. 3. Find the critical values for the F-distribution using the F-table. For \(\alpha = 0.05\) and df1 = df2 = 19, the critical values are F(0.025, 19, 19) = 2.34 and F(0.975, 19, 19) = 0.43. 4. Compare the F statistic to the critical values. If the F statistic falls within the critical values range, we fail to reject the null hypothesis. If not, we reject the null hypothesis. Since 1.21 is between the critical values 0.43 and 2.34, we fail to reject the null hypothesis. Therefore, we can assume that the population variances are equal.

Part b: Comparing means with appropriate test

Since we concluded that variances of two populations are equal, we will use the two-sample t-test to compare the population means. The null hypothesis (H0) is that there is no difference between the average amount spent by the two groups, and the alternative hypothesis (H1) is that there is a difference between the average amount spent by the two groups. - Null hypothesis (H0): \(\mu_1 = \mu_2\) - Alternative hypothesis (H1): \(\mu_1 ≠ \mu_2\) Using the given samples and the significance level α=0.05: 1. Compute the pooled variance: $$S_p^2 = \frac{(n_1 - 1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2} = \frac{(19)(22^2) + (19)(20^2)}{38} = \frac{18436}{38} = 485.42$$ 2. Compute the standard error of the difference between means: $$SE(\overline{X}_1 - \overline{X}_2) = \sqrt{\frac{S_p^2}{n_1} + \frac{S_p^2}{n_2}} = \sqrt{\frac{485.42}{20} + \frac{485.42}{20}} = 10.33$$. 3. Calculate the t statistic: $$t = \frac{(\overline{X}_1 - \overline{X}_2)}{SE(\overline{X}_1 - \overline{X}_2)} = \frac{(78 - 67)}{10.33} = 1.065$$. 4. Determine the degrees of freedom for the t-test: df = n1 + n2 - 2 = 38. 5. Find the critical t-values using the t-table for a two-tailed test with \(\alpha=0.05\) and df = 38: t(0.025, 38) = 2.024 and t(0.975, 38) = -2.024. 6. Compare the t statistic to the critical t-values. If the t statistic falls within the critical values range, we fail to reject the null hypothesis. If not, we reject the null hypothesis. Since 1.065 is between -2.024 and 2.024, we fail to reject the null hypothesis. Therefore, there is no significant difference between the average amount spent per trip on weekends versus weekdays at the α=0.05 level of significance.

Key concepts

Understanding Variances Equality

When comparing two different groups, like weekend and weekday shoppers, we often need to examine if their variances are the same. Variability, or variance, tells us how spread out the spending amounts are within each group. This concept of variances equality tests whether the differences in variability are significant or just due to random sampling.

To test this, we use the F-test, which compares the ratio of the two variances. In our example, we calculated this ratio to be 1.21. We then compared it to critical values from the F-distribution table. If the value of 1.21 falls within the range of these critical values (0.43 to 2.34), we decide the variances are equal. Here, they are equal, so we don't see a significant difference in variance between weekend and weekday spending habits.

The F-test is crucial in hypothesis testing to ensure we're using correct analyses to compare group means later. Ensuring variances equality keeps our comparisons valid and reliable.

T-Test: A Tool for Comparing Means

The t-test is a statistical method used to compare the means of two groups and see if they are statistically different from each other. Since we determined that the variances are equal, a two-sample t-test is appropriate for our data on shopper spending.

The null hypothesis in this t-test states that there is no difference in the average spending between weekend and weekday shoppers (\( \mu_1 = \mu_2 \)). The alternative hypothesis suggests a difference exists (\( \mu_1 eq \mu_2 \)).

Using our sample data, we calculate a pooled variance, which helps us find the standard error of the difference in means. This pooled variance considers both groups and their size. We then compute the t-statistic by dividing the difference in sample means by this standard error. For our data, the t-statistic is 1.065, lying between the critical values -2.024 and 2.024. This result tells us there's not enough evidence to conclude a significant difference.

The t-test shows whether differences are meaningful or due to chance, guiding informed conclusions about data.

Exploring Hypothesis Testing

Hypothesis testing is a structured method to make decisions based on data, like deciding if there's a difference in spending habits between different shopper groups. It begins with formulating two hypotheses: the null hypothesis (\( H_0 \)) assumes no effect or change, while the alternative hypothesis (\( H_1 \)) suggests there is a difference.

Steps in hypothesis testing include:

• Choosing a significance level (\( \alpha \)), often set at 0.05; it represents a 5% risk of concluding a difference exists when there is none.
• Calculating a relevant statistic (F or t in our context) from the data.
• Comparing this statistic against critical values from statistical distribution tables.
• Making a decision: if the statistic lies within the critical range, we fail to reject the null hypothesis. Otherwise, we reject it.

In this process, understanding and evaluating assumptions, like equal variances, ensure the test results' accuracy and trustworthiness. Hypothesis testing allows us to draw conclusions about population-level effects based on sample data, providing an essential tool in statistics and research.