Question

Trying To Establish Whether Men And Women Value The Product Differently. They Have Data On The Willingness To Pay For The Product For 50 Men And 50 Women. Design A Hypothesis Test Of Whether Or Not The Mean Willingness To Pay For The Product Is Different For Men And Women. Which, If Any, Is More

A company is looking at the results of a marketing survey and trying to establish whether men and women value the product differently. They have data on the willingness to pay for the product for 50 men and 50 women.

Design a hypothesis test of whether or not the mean willingness to pay for the product is different for men and women.

Which, if any, is more important to minimize in this case, the type I or type II error? Why?

How does the choice of the critical value affect the tradeoff between the type I and type II error?

Answer 1

To test if there is a significant difference in the mean willingness to pay between men and women, one should formulate null and alternative hypotheses, use appropriate statistical distributions, calculate the test statistic, and examine the results considering the critical value. The choice of the critical value affects the trade-off between Type I and Type II errors, which should be considered based on the context and consequences.

Step-by-step explanation:

To design a hypothesis test to determine if there is a difference in the mean willingness to pay for a product between men and women, the following steps can be followed:

1. Define the null hypothesis (H0): The means are equal, so H0: μmen = μwomen.
2. Define the alternative hypothesis (Ha): The means are not equal, so Ha: μmen ≠ μwomen.
3. In words, the random variable is the difference in mean willingness to pay for the product.
4. The distribution to use is likely a t-distribution if the standard deviations are unknown and the sample size is small, or a z-distribution if standard deviations are known and/or sample size is large.
5. Calculate the test statistic using sample means, standard deviations, and sample sizes of both groups.
6. Create a graph of the distribution and shade the area representing the actual level of significance which corresponds to the critical value(s). If the test is two-tailed, shade both tails; if one-tailed, shade the direction indicated by Ha.

As for which type of error is more important to minimize, it often depends on the context. In a marketing survey like this, a Type I error (rejecting a true null hypothesis) might lead to wrong marketing strategies that could be costly. A Type II error (failing to reject a false null hypothesis) might result in missing out on a potential market difference that could be exploited. The decision should be based on the cost and consequences of each error.

The choice of the critical value affects the tradeoff between Type I and Type II errors. A larger critical value (such as 0.05 compared to 0.01) increases the chance of a Type I error but decreases the chance of a Type II error, and vice versa.