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Change coffee brand? A study was conducted to see if an advertisement campaign would increase market share for Sanka instant decaffeinated coffee (R. Grover and \(\mathrm{V}\). Srinivasan, \(J .\) Marketing Research, vol. \(24,1987,\) pp. 139 - 153). Subjects who use instant decaffeinated coffee were asked which brand they bought last. They were asked this before the campaign and after it. The results are shown in the table, with computer output based on it below. a. Estimate the population proportion choosing Sanka for the (i) first purchase and (ii) second purchase. Find the difference and interpret it. b. Explain how to interpret the \(95 \%\) confidence interval. c. The software output also shows a P-value. State the hypotheses that this P-value refers to in the context of this exercise and give the conclusion of the test when using a significance level of \(0.05 .\) \begin{tabular}{lcc} \hline Two Purchases of Coffee & \\ \hline & \multicolumn{2}{c} { Second Purchase } \\ \cline { 2 - 3 } First Purchase & Sanka & Other Brand \\ \hline Sanka & 155 & 49 \\ Other brand & 76 & 261 \\ \hline \end{tabular}

Step by step solution

01

Calculate Proportion for First Purchase

To estimate the proportion of the population choosing Sanka in the first purchase, we add the total number of people who bought Sanka in the first purchase, which is 155. The total number of respondents for the first purchase is the sum of everyone who bought Sanka or other brands, calculated as 155 + 49 + 76 + 261 = 541. Thus, the proportion is calculated by dividing the number who bought Sanka by the total respondents: \( p_1 = \frac{155}{541} \approx 0.286 \text{ (rounded to three decimal places)} \).

02

Calculate Proportion for Second Purchase

Now, we calculate the proportion of the population choosing Sanka in the second purchase. The total number of respondents who bought Sanka in the second purchase is 155 + 76 = 231. The same total number of respondents, 541, applies to the second purchase. Thus, the proportion is \( p_2 = \frac{231}{541} \approx 0.427 \text{ (rounded to three decimal places)} \).

03

Compute and Interpret the Difference in Proportions

The difference in proportions, \( p_2 - p_1 \), represents the change in the proportion of the population choosing Sanka between the first and second purchases. Calculating the difference gives \( 0.427 - 0.286 = 0.141 \). This indicates a 14.1% increase in the proportion of customers choosing Sanka after the campaign.

04

Interpret the 95% Confidence Interval

The 95% confidence interval provides a range of values which we are 95% confident contains the true difference in proportions for the entire population. It considers the variability in the sample data. If it does not contain zero, it suggests a statistically significant change in the proportion of people purchasing Sanka.

05

State and Conclude the Hypothesis Test

The null hypothesis (\(H_0\)) usually states that there is no difference in proportions before and after the campaign: \( p_1 = p_2 \). The alternative hypothesis (\(H_a\)) states there is a difference: \( p_1 eq p_2 \). Given a P-value less than 0.05, we reject \(H_0\) in favor of \(H_a\), concluding a significant increase in Sanka's market share post-campaign.

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Key Concepts

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

Market Share Analysis

Market share analysis is critical for understanding how a product or brand performs in relation to its competitors within an industry. In the given study about Sanka coffee, market share analysis was used to assess the impact of an advertising campaign on consumer behavior. By comparing the number of customers who purchased Sanka before and after the promotional effort, we can determine the campaign’s effectiveness.

When conducting a market share analysis, it is essential to look at:

• The proportion of customers buying a specific brand.
• The changes in these proportions over time or following interventions, like marketing campaigns.

In this exercise, you calculated the proportion of Sanka purchases at two different points – before and after the advertising campaign. By evaluating these changes, companies can make informed decisions about marketing strategies and investments.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. In the context of the Sanka coffee study, hypothesis testing determines whether the observed difference in market share pre- and post-campaign is statistically significant.

Here’s how the hypothesis testing process works:

• The null hypothesis ( \(H_0\) ): This proposition suggests no difference between the two proportions, meaning the campaign had no effect.
• The alternative hypothesis ( \(H_a\) ): This asserts there is a difference, indicating that the campaign did impact market share.
• The P-value: This probability value helps us decide whether to reject the null hypothesis. If the P-value is below a chosen significance level (often 0.05), it indicates that there is enough evidence to accept the alternative hypothesis.

By applying hypothesis testing in this scenario, we can conclude if Sanka's marketing campaign indeed had a significant influence on its market share.

Confidence Intervals

Confidence intervals provide a range of values that are believed to encompass the true population parameter with a specified level of certainty, typically 95%. In the Sanka coffee case, we calculate a confidence interval for the difference in proportions of people buying Sanka before and after the campaign.

Why is the confidence interval valuable?

• It gives a range in which the true difference in proportions likely lies.
• It allows us to ascertain the reliability of our estimates.
• If the interval doesn’t include zero, it suggests a statistically significant difference, supporting the alternative hypothesis.

The confidence interval helps us understand not only the estimated change but also how precise this estimate is. Thus, it is an integral component of interpreting survey results and making economic decisions.

Proportions

Proportions are used in statistics to describe the size of one part relative to the whole. In our coffee study, proportions help us to understand what fraction of the respondents chose Sanka coffee compared to all survey participants.

Calculating proportions is straightforward:

• Determine the part of interest (e.g., customers choosing Sanka).
• Divide this number by the total number of survey participants.

For example, in the initial purchase phase, the proportion is calculated as the number of people who bought Sanka divided by the total number of respondents. This provides a metric reflecting consumer preferences at a given time.

Understanding proportions helps businesses and researchers gauge the size of their market presence and can be crucial in strategic planning and competitive analysis.