A Producer Of Fine Chocolates Believes That Sales Of Two

A producer of fine chocolates believes that the sales of two varieties

A producer of fine chocolates believes that the sales of two varieties of truffles differ significantly during the holiday season. The first variety is milk chocolate, while the second is milk chocolate filled with mint. It is assumed that the sales of truffles are normally distributed with unknown but equal population variances. Two independent samples of 18 observations each were collected for the holiday period. The sample mean of milk chocolate truffles sold is 12 million with a sample standard deviation of 2.5 million, and the sample mean of mint-filled chocolate truffles sold is 13.5 million with a sample standard deviation of 2.3 million. Using milk chocolate as population 1 and mint chocolate as population 2, the appropriate hypotheses to determine if the average sales of milk chocolate truffles are at least the same as the sales of mint chocolate truffles are:

Paper For Above instruction

When comparing the sales of two different chocolate varieties, specifically milk chocolate and mint-filled chocolate truffles, formulating the correct hypotheses for statistical testing is crucial for accurate inference. In this context, the producer wants to determine whether the average sales of milk chocolate truffles are at least equal to or greater than those of mint chocolate truffles during the holiday season. This corresponds to a hypothesis test comparing two population means, under the assumption that the sales data are normally distributed with equal variances.

The relevant hypotheses involve defining the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis generally states the status quo or no difference, while the alternative hypothesis reflects the specific difference or inequality the producer wants to test. For this scenario, the key question is whether the mean sales for milk chocolate are at least as high as those for mint chocolate. In statistical terms, this is a one-sided test where the null suggests that milk chocolate sales are less than the mint chocolate sales, or at best equal.

The options provided include various formulations involving the population means (μ1 for milk chocolate and μ2 for mint chocolate) and their differences. Evaluating each:

- Option (A): H0: μ1 - μ2 ≥ 0; H1: μ1 - μ2

This test examines whether the difference in sales (milk minus mint) is less than zero, implying milk sales are less than mint sales. Since the producer is interested in whether milk sales are at least equal to mint sales, this is the opposite of the hypothesis needed.

- Option (B): H0: μD = 0; H1: μD ≠ 0

Here, μD refers to the difference in means (μ1 - μ2). This is a two-tailed test assessing whether the means differ, without specifying direction. However, the producer specifically wants to test if milk sales are at least as high, which typically involves a one-sided test.

- Option (C): H0: μD ≥ 0; H1: μD

This tests whether the difference (milk minus mint) is less than zero, which is inconsistent with testing if milk sales are at least as high as mint sales.

- Option (D): H0: μ1 - μ2 = 0; H1: μ1 - μ2 ≠ 0

This is a two-sided test for no difference, which does not align with the goal of testing whether milk sales are at least as high as mint sales.

Based on the above, the proper hypotheses for testing whether the average sales of milk chocolate truffles are at least equal to those of mint chocolate truffles are: The null hypothesis that the difference is greater than or equal to zero (H0: μ1 - μ2 ≥ 0), indicating that milk sales are at least as high as mint sales, against the alternative that this difference is less than zero, meaning milk sales are lower.

Hence, option (A) aligns with this reasoning, providing a one-sided test for whether milk chocolate sales are less than mint sales. However, if the primary interest is to test whether milk sales are at least as high as mint, the hypotheses should be reversed to reflect testing for the difference being less than or equal to zero (or equivalently, the null being that the difference is ≥ 0, and the alternative being that it is

Final selection: Option (A) is appropriate for testing whether the sales of milk chocolate are less than the mint chocolate sales—useful if the goal is to see if milk sales are not at least as high. But for confirming whether milk sales are at least as high, the hypotheses should be:

- Null hypothesis: μ1 - μ2 ≥ 0 (milk sales are at least equal to mint sales)

- Alternative hypothesis: μ1 - μ2

Therefore, option (A) correctly models the hypotheses when testing whether milk sales are less than mint sales, aligning with the assertion "at least the same" if phrased as a test for not being less. Alternatively, a more precise formulation for "at least the same" could be H0: μ1 - μ2 ≥ 0 versus H1: μ1 - μ2

In conclusion, the most appropriate hypotheses for testing if the average sales of milk chocolate are at least the same as those of mint chocolate during the holiday season are:

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