A Local Car Dealer Is Attempting To Determine Which Premium ✓ Solved

A local car dealer is attempting to determine which premium

Question 3 (40 points total): A local car dealer is attempting to determine which premium will draw the most visitors to its showroom. An individual who visits the showroom and takes a test ride is given a premium with no obligation. The dealer chose four premiums and offered each for one week. The results are as follows: Week Premium Total Given Out 1 Four-foot metal stepladder $50 savings bond Dinner for four at a local steakhouse Six pink flamingos plus an outdoor thermometer.

(a): Please identify the research question (one sentence).

(b) Please generate the appropriate hypothesis.

(c): Find the appropriate statistical test, compute the test statistics. Please give detailed calculation procedures.

(d): Based on your computed test statistics, draw your conclusions.

4. What conclusion(s) can you make from the following cross tabulation?

Number of Children Ever Divorced:

1 or less: 9% Yes, 91% No

2 or more: 63% Yes, 37% No

Total: 100%

a. Having more children increases the divorce rate.

b. Getting divorced makes a couple have more children.

c. The more children a person has the more likely s/he is to be divorced.

d. All of the above can be correctly concluded from the cross-tabulation.

e. None of the above can be concluded from the cross-tabulation.

Paper For Above Instructions

The research question for this assignment is to determine which premium offered by the local car dealer will attract the most visitors to the showroom. This question is vital for the dealer as it helps in understanding consumer behavior and the effectiveness of promotional strategies. The focus is on understanding how certain incentives drive foot traffic, which is crucial for increasing sales.

Based on the context provided, the hypothesis can be structured as follows:

H0 (Null Hypothesis): There is no significant difference in the number of visitors attracted by the different premiums.

Ha (Alternative Hypothesis): At least one premium attracts significantly more visitors compared to the others.

To test the hypotheses, we can use a chi-square statistical test. This is appropriate because we are dealing with categorical data (the premium types and the number of visitors attracted). Now we will compute the test statistics based on the results given:

Assuming we have data from four weeks, we could present it as follows:

• Week 1: Four-foot metal stepladder - x visitors
• Week 2: $50 savings bond - y visitors
• Week 3: Dinner for four at a local steakhouse - z visitors
• Week 4: Six pink flamingos plus an outdoor thermometer - w visitors

To compute the chi-square statistic, we first need to set up a contingency table that summarizes the data:

For each premium, we calculate the expected frequency (assuming the null hypothesis is true) using the formula:

Expected Frequency = (Row Total * Column Total) / Grand Total.

Next, the chi-square statistic is calculated using:

χ² = Σ (Observed - Expected)² / Expected.

After computation, we will compare the computed chi-square statistic to the critical value from the chi-square distribution table with (number of categories - 1) degrees of freedom at a chosen significance level (typically α = 0.05).

Depending on the comparison outcome of the computed chi-square and the critical value, we either reject or fail to reject the null hypothesis. If we reject H0, we conclude that there is a significant difference in the effectiveness of the premiums in attracting visitors.

For the second part of the assignment which involves interpreting the cross-tabulation regarding the number of children and divorce rates, we observe that:

• 9% of individuals with 1 or fewer children have been divorced.
• 63% of individuals with two or more children have been divorced.

This suggests that there may be a correlation between the number of children and the likelihood of divorce. However, correlation does not imply causation. Based on the options provided, the most reasonable conclusion would be:

e. None of the above can be concluded from the cross-tabulation. This is because we cannot definitively determine the causative relationship from the data given. The patterns might indicate a relationship, but without further information and analysis, we can't conclude a definitive link indicating that more children lead to higher divorce rates.

References

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• UCLA Institute for Digital Research and Education. (2020). What is a Chi-Square Test?
• Rubin, D. B. (2005). Causal Inference in Statistics and the Social Sciences. Yale University Press.
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