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How to sell a burger A fast-food chain wants to compare two ways of promoting a new burger (a turkey burger). One way uses a coupon available in the store. The other way uses a poster display outside the store. Before the promotion, their marketing research group matches 50 pairs of stores. Each pair has two stores with similar sales volume and customer demographics. The store in a pair that uses coupons is randomly chosen, and after a monthlong promotion, the increases in sales of the turkey burger are compared for the two stores. The increase was higher for 28 stores using coupons and higher for 22 stores using the poster. Is this strong evidence to support the coupon approach, or could this outcome be explained by chance? Answer by performing all five steps of a two-sided significance test about the population proportion of times the sales would be higher with the coupon promotion.

Step by step solution

01

State the Hypotheses

We need to set up our null and alternative hypotheses. The null hypothesis (\(H_0\)) states that there is no difference in effectiveness between using coupons and posters, meaning the probability (\

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. The process helps determine whether there is enough evidence to reject a null hypothesis, which often claims that there is no effect or no difference between groups.
Here are the main components:

• Null Hypothesis (87): Typically a statement of no effect or no difference, it serves as the default assumption. In this case, the null hypothesis would be that the use of coupons is equally effective as posters in increasing sales of the turkey burger.
• Alternative Hypothesis (88): This is what the researcher wants to prove. In the burger promotion scenario, the alternative hypothesis would propose that coupons are more effective than posters in boosting sales.
• Significance Level (19): Often set at 0.05, it's the threshold for determining whether an observed effect is statistically significant.
• Test Statistic: A numerical value calculated from the sample data, compared against a critical value to decide whether to reject the null hypothesis.
• Decision Rule: If the test statistic exceeds the critical value, the null hypothesis is rejected in favor of the alternative hypothesis.

Understanding hypothesis testing is essential as it helps companies like the fast-food chain make informed decisions about promotional strategies.

Population Proportion

Population proportion is a statistical measure that represents the fraction of the population with a particular characteristic. In the context of the sales promotion analysis, the population proportion is the probability that the coupon strategy results in a higher increase in burger sales compared to the poster strategy.
For hypothesis testing of population proportions, we often compare the observed proportion from the sample with a known proportion or hypothesized proportion based on the null hypothesis.
In our example, the observed result shows that out of the 50 pairs of stores, 28 used the coupon strategy effectively, giving us an observed proportion (7_hat) of 7_hat = 28/50 = 0.56.

• Null Hypothesis (87): Here, the hypothesized population proportion (P) might be 0.5, suggesting equal effectiveness between coupons and posters.
• Alternative Hypothesis (88): This states that the population proportion is greater than 0.5, supporting that the coupon strategy is more effective.
• Z-test for Proportion: Used to assess whether the observed proportion significantly deviates from the hypothesized proportion.

By analyzing these proportions, the company can assess the relative success of the coupon promotion strategy against the poster strategy.

Sales Promotion Analysis

Sales promotion analysis involves examining the effects of different marketing strategies on sales figures. It helps businesses understand what works best to enhance their revenue streams.
For this particular exercise, the fast-food chain is evaluating whether coupons or poster displays are more effective at increasing burger sales.

• Matched Pairs Design: Stores are paired based on similar characteristics to limit variability that could skew results, ensuring a fair comparison between the two promotional efforts.
• Significance of Differences: By comparing sales increases between matched pairs, the chain can determine if coupons outperform posters beyond what might be expected by chance.
• Quantitative Analysis: Examining numerical differences in sales provides concrete data on promotional effectiveness, facilitating data-driven decision-making.

The results of the analysis offer actionable insights, guiding the chain's future marketing strategies and aiding in the efficient allocation of promotional resources.