Question

Taste Testing In a head-to-head taste test of storebrand foods versus national brands, Consumer Reports found that it was hard to tell the difference. \({ }^{4}\) If the national brand is indeed better than the store brand, it should be judged as better more than \(50 \%\) of the time. a. State the null and alternative hypotheses to be tested. Is this a one- or a two-tailed test? b. Suppose that, of the 25 food categories used for the taste test, the national brand was found to be better than the store brand in 7 of the taste comparisons, while in 10 pairs, the tasters could taste no difference between the two. Use the sign test to test the hypothesis in part a with \(\alpha \approx .05 .\) What practical conclusions can you draw?

Short answer

Answer: No, based on the taste test results, there is not a significant difference in taste that favors the national brand over the store brand.

Step-by-step solution

State the null and alternative hypotheses

The null hypothesis (H0) states that the national brand is not better than the store brand in terms of taste, which means it would be judged as better than the store brand 50% of the time. The alternative hypothesis (Ha) states that the national brand is better than the store brand in terms of taste so that it would be judged as better more than 50% of the time: - H0: p = 0.50 - Ha: p > 0.50 Since we are only interested in whether the national brand is judged to be better more than 50% of the time, this is a one-tailed test.

Perform the Sign Test

In this step, we have a total of 25 food categories. The national brand was found to be better than the store brand in 7 of the taste comparisons, while in 10 pairs, there was no difference in taste between the two. The sign test is a non-parametric test that only considers the signs of the differences between pairs (ignoring cases where the difference is zero), so we will not take into account the 10 pairs where the tasters could taste no difference. Now, we have 7 positive differences (national brand better) and 25 - 10 - 7 = 8 negative differences (store brand better). In this case, n = 15 (ignoring the 10 ties). We want to test the hypothesis with α ≈ 0.05. The sign test examines the probability of getting 7 or fewer positive differences (given the null hypothesis is true) from a binomial distribution: - P(x ≤ 7) = P(x=0) + P(x=1) + ... + P(x=7) Since this is a one-tailed test, we will compare the cumulative probability of getting x ≤ 7 with α (significance level) ≈ 0.05: - If P(x ≤ 7) ≤ α, reject H0 (accept Ha) - If P(x ≤ 7) > α, fail to reject H0 We can use either a binomial table or a calculator to find the probability. Let's use the formula: P(X = k) = (nCk) * (p ^ k) * (1 - p) ^ (n - k) where nCk is the number of combinations of choosing k positive differences from n differences (n! / (k! * (n-k)!)). After calculating, we get P(x ≤ 7) ≈ 0.6585. Since 0.6585 > 0.05, we fail to reject the null hypothesis.

Draw Practical Conclusions

Since we failed to reject the null hypothesis (p = 0.50), we cannot conclude that the national brand is significantly better in terms of taste compared to the store brand. Therefore, based on this taste test, there isn't substantial evidence to support the claim that the national brand is better in taste compared to the store brand. Consumers may not necessarily prefer the national brand over the store brand based on taste alone.

Key concepts

Sign Test

The Sign Test is a simple non-parametric test used to evaluate a hypothesis about the median of a single sample or paired samples. In situations like taste tests, where the preference for one option over another can be categorized as better, worse, or no difference, the sign test is useful. It ignores the magnitude of the difference and instead focuses on the direction of differences (positive or negative).
In our example, we counted how often the national brand was deemed better than the store brand. Out of 25 comparisons, 10 had no difference, leaving us with 15 comparisons. Here are the steps:

• Count how many times the national brand was better (positive sign).
• Count how many times the store brand was better (negative sign).

In this test, the 7 times the national brand was better are positive differences, and the remaining 8 times are negative differences. We compare these counts against a binomial distribution to decide if there is enough evidence to support the claimed superiority of the national brand.

Null Hypothesis

The Null Hypothesis (denoted as \(H_0\)) is a foundational concept in hypothesis testing. It represents the default position that there is no effect or difference. Essentially, it is a statement of 'no change' or 'no difference.' In our taste test scenario, the null hypothesis would suggest that there is no significant difference in taste preference between the national and store brands.
For our example, the Null Hypothesis is set as follows:

• \(H_0: p = 0.50\): The probability that the national brand is selected over the store brand is 50%.

This implies that when people choose between the brands, they are doing so randomly, showing no real preference for the national brand. The aim of our hypothesis test is to determine if we have enough statistical evidence to reject this assumption.

Alternative Hypothesis

The Alternative Hypothesis (denoted as \(H_a\)) suggests that there is a significant difference or effect present. Unlike the null hypothesis, it is what you might hope to demonstrate as true through your test. In our exercise, it asserts that the national brand is more likely to be chosen based on taste in comparison to the store brand.
The alternative hypothesis is often what motivates the research or test. In terms of the taste test:

• \(H_a: p > 0.50\): The probability that the national brand is selected over the store brand is greater than 50%.

This hypothesis leads us to conduct a one-tailed test, as we are only interested in detecting an improvement in preference for the national brand, not a decrease or stability. The goal is to see if the evidence strongly supports an above-random preference for the national brand.

Binomial Distribution

The Binomial Distribution is a probability distribution that summarizes the likelihood of a value occurring in an experiment that has two possible outcomes, often termed 'success' or 'failure.' In our taste test, the success could be the national brand being preferred, and failure would be the store brand being preferred. We use the binomial distribution to model this scenario, as each choice can be categorized as one or the other.
Key features of a binomial distribution involve:

• Number of trials (n), in our case, 15 trials (tastings)
• Probability of success (p), here assumed to be 0.50 under the null hypothesis
• Number of successes (x), here the 7 times the national brand was preferred

By computing the probability of achieving 7 or fewer successes under these conditions, we assess whether such a result would be unusual if there really were no preference for the national brand. This helps us decide whether to reject the null hypothesis.