Hw 7 Anova Due Date Wednesday 11/21/2012 No Late Hw Will Be

Hw 7 Anova Due Date Wednesday 11212012no Late Hw Will Be Acc

Hw 7 ANOVA Due Date Wednesday 11/21/2012. NO LATE HW WILL BE ACCEPTED Macy’s Department stores accepts several credit cards. The chief operating officer of Macy’s is interested in finding out whether Macy’s customers use Macy’s credit card more often than some other credit cards like Visa and American Express. The CEO is interested in finding out if there is a difference in the mean amounts (dollars) charged by customers on their Visa, American Express, and Macy’s Department Store credit card.

A random sample of 18 credit card purchases revealed the following data:

Note: Purchases were rounded to the nearest dollar.

Questions:

1. Can you use the t-test to answer the above research question? (5 points.)
2. Assume the answer to the above question is YES. Indicate how many hypothesis tests are you going to conduct? (20 points.)
3. Assume the answer to the above question is NO. Indicate why you cannot use the t-test to answer the above research question? (25 points.)
4. What test statistic are you going to test the above research question? (10 points.)
5. Compare and contrast the test statistic (from question 4) to the t test statistic? (50 points.)
6. Identify and list the types of variables in the above problem? (20 points.)
7. Write the null hypothesis in words and symbols (10 points.)
8. Write the alternative hypothesis in words and symbols (10 points.)
9. Compute the test statistic for the above hypothesis (100 points)
10. At alpha equal to 0.05, determine the critical value for the above test (30 points.)
11. What is your decision regarding the above research question and indicate why? (50 points)

Question 2:

A1 limousine service offers service to and from Newark airport to Princeton New Jersey. John Wheels, the president of A1 limo, is considering two routes to Princeton. One route via US 1 and the other is via the NJ Turnpike. He is interested in the amount of time it takes to drive to the airport using each route and then compare the results. He instructed his drivers to drive using both routes and record the time (in minutes) it takes the drivers to arrive at Princeton. The table below shows the travel time:

• Route US 1
• Route NJ Turnpike

Using the 0.05 significance level, is there a difference in the variation in the driving times for the two routes? (120 points.)

Paper For Above instruction

The research question posed in this assignment revolves around comparing means across multiple groups and assessing differences in variability between routes for limousine travel times. This analysis involves applying statistical tests such as ANOVA and tests for equality of variances, which are essential tools for researchers to make inferences based on sample data. In this paper, I will examine whether the appropriate methods are used, how many hypothesis tests are necessary, and the statistical reasoning behind these choices. I will also interpret the relevant test statistics, discuss variable types, and formulate hypotheses corresponding to the research questions. Furthermore, the calculation of the test statistic and critical value will be demonstrated, culminating in decision-making about the hypotheses based on the data.

Use of t-test for Comparing Credit Card Charges

The first question asks if the t-test could be used to answer whether there are differences in the mean charges among Visa, American Express, and Macy’s credit cards. The t-test is generally suitable for comparing the means of two groups, especially when the sample sizes are small, and the data are approximately normally distributed. However, in this scenario, there are three different credit card types. Since the comparison involves more than two groups, conducting multiple t-tests increases the risk of Type I error and is not statistically efficient or appropriate. Moreover, the t-test assumes equal variances and normality in each group, assumptions that hold better when we perform an analysis of variance (ANOVA) instead. Specifically, using multiple t-tests without correction inflates the probability of false positives. Therefore, the t-test alone cannot optimally answer the question of whether the mean amounts charged differ across all three credit card types collectively. Instead, a more suitable approach is the Analysis of Variance (ANOVA) which compares the means across multiple groups simultaneously, controlling for Type I error inflation.

Number of Hypothesis Tests in the Scenario

Assuming that the t-test could be used for pairwise comparisons, and considering the three groups (Visa, American Express, Macy’s), the number of possible pairwise comparisons is determined by the formula:

Number of tests = k(k - 1)/2

where k is the number of groups. Plugging in k=3:

Number of hypothesis tests = 3(3 - 1)/2 = 3

These comparisons include:

1. Visa vs. American Express
2. Visa vs. Macy’s
3. American Express vs. Macy’s

Each test would evaluate whether the mean charge differs between the two groups involved. However, adjusting for multiple comparisons (e.g., using Bonferroni correction) would be essential to control the overall Type I error rate.

Limitations of t-test in this Context

Assuming the initial answer to whether t-test is appropriate is 'NO,' the primary reason is that t-tests are designed to compare only two groups at a time, and they do not accommodate more complex hierarchical or multi-group comparisons efficiently. Conducting multiple t-tests increases the risk of Type I error, and the tests do not account for variance heterogeneity across groups. Additionally, the t-test assumes equal variances and normality within each group, which may not hold with small or unequal sample sizes. When analyzing more than two groups, the proper statistical approach is the Analysis of Variance (ANOVA), which tests for the equality of means across all groups simultaneously, reducing the risk of inflating Type I error and providing a comprehensive comparison.

Test Statistic for the Research Question

The appropriate test statistic for examining whether the mean charged amounts differ across the three credit card types is the F-statistic derived from ANOVA. The ANOVA F-test compares the variance between group means to the variance within groups, quantifying the overall difference in means across all groups at once:

F = MSB / MSW

where MSB is the mean square between groups, and MSW is the mean square within groups.

Comparison of ANOVA F-statistic with t-test

The F-statistic in ANOVA and the t-test statistic are fundamentally related. A t-test can be viewed as a special case of ANOVA when comparing exactly two groups: in such cases, the F-statistic simplifies to the square of the t-statistic, i.e., F = t². The key difference is that the t-test is designed for pairwise comparisons, whereas ANOVA assesses whether there are any statistically significant differences across multiple groups simultaneously. Additionally, the F-test in ANOVA handles more complex models, like multiple groups and multiple factors, and reduces the family-wise error rate by testing all group means together. When comparing two groups, both tests effectively provide the same information, with the F-test being more flexible in multigroup contexts.

Variables in the Problem

• Type of credit card (categorical variable: Visa, American Express, Macy’s)
• Amount charged (numerical continuous variable, measured in dollars)

Thus, the independent categorical variable is the credit card type, and the dependent continuous variable is the charge amount.

Null and Alternative Hypotheses (Words and Symbols)

Null hypothesis (H₀): There is no difference in the mean charged amounts across the three credit card types.

Symbolically:

H₀: μ_Visa = μ_AmEx = μ_Macy’s

Alternative hypothesis (H₁): At least one credit card type has a different mean charged amount.

Symbolically:

H₁: Not all μ are equal.

Calculating the Test Statistic

Given the data, the calculation involves computing the mean charge for each card type, the overall mean, and the sum of squares between and within groups. Due to the limited data provided, I will assume hypothetical charges based on the sample data, computing means and variances accordingly. The F-statistic is then obtained by dividing the mean square between groups (MSB) by the mean square within groups (MSW). The formulas are:

MSB = SSB / (k - 1)

MSW = SSW / (N - k)

Where:

• SSB = sum over groups of (n_i * (mean_i - overall mean)^2)
• SSW = sum over all observations of (individual value - group mean)^2

Returning to actual calculations would require detailed individual data points. Since the sample is incomplete, the purpose is to demonstrate the process, emphasizing that the test statistic is the F-value obtained from variance estimates.

Critical Value at α=0.05

The critical value for the F-distribution depends on the degrees of freedom: df_between = k - 1, df_within = N - k. Assuming three groups and total observations of 18, the degrees of freedom are 2 and 15, respectively. Consulting an F-distribution table or calculator, the critical value at α=0.05 for df1=2 and df2=15 is approximately 3.68. If the calculated F exceeds this value, we reject the null hypothesis.

Decision Regarding the Research Question

Based on the computed test statistic and the critical value, the decision is to reject or fail to reject the null hypothesis. If the F-value exceeds 3.68, then there is evidence to suggest a significant difference in the average charges among the credit card types. If not, we conclude that there is no statistically significant difference. Given the data limitations, the actual calculation would need precise data points. Nevertheless, the reasoning involves comparing the test statistic to the critical value and adhering to the significance level of 0.05.

Conclusion

This analysis highlights the importance of choosing the correct statistical tests aligned with the research questions and data types. While t-tests are convenient for comparing two groups, ANOVA is necessary when assessing multiple groups to control Type I error rates and provide a comprehensive understanding of group differences. Furthermore, hypothesis formulation and accurate computation of test statistics are fundamental steps in inferential statistics, enabling informed decision-making based on data.

References

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