Question

Two proposed computer mouse designs were compared recording wrist extension in degrees for 24 people who each used both mouse types ("Comparative Study of Two Computer Mouse Designs" Cornell Human Factors Laboratory Technical Report RP7992). The difference in wrist extension was computed by subtracting extension for mouse type \(\mathrm{B}\) from the wrist extension for mouse type \(\mathrm{A}\) for each student. The mean difference was reported to be \(8.82\) degrees. Assume that it is reasonable to regard this sample of 24 people as representative of the population of computer users. a. Suppose that the standard deviation of the differences was 10 degrees. Is there convincing evidence that the mean wrist extension for mouse type \(\mathrm{A}\) is greater than for mouse type \(\mathrm{B}\) ? Use a \(.05\) significance level. b. Suppose that the standard deviation of the differences was 25 degrees. Is there convincing evidence that the mean wrist extension for mouse type \(\mathrm{A}\) is greater than for mouse type B? Use a .05 significance level. c. Briefly explain why a different conclusion was reached in the hypothesis tests of Parts (a) and (b).

Short answer

The conclusion on whether mouse type A causes more wrist extension than type B depends on the standard deviation. A smaller standard deviation, such as 10 degrees, might provide enough evidence to reject the null hypothesis at .05 significance level, implying that we have evidence to suggest mouse type A does cause more wrist extension. However, if the standard deviation is larger, such as 25 degrees, we might not have enough evidence to reject the null hypothesis at this significance level, implying that we cannot suggest that mouse type A causes more wrist extension.

Step-by-step solution

Set up the Hypotheses

First, we need to establish the null hypothesis (H0) and the alternative hypothesis (Ha). Our H0 is that there is no difference in the mean wrist extension for mouse types A and B, i.e., \(\mu = 0\). On the other hand, our Ha is that the mean wrist extension for mouse type A is greater than for mouse type B, i.e., \(\mu > 0\).

Calculate Test Statistics for each standard deviation

Next, calculate the test statistic. The formula for this in our case, where we're given the standard deviation, mean difference, and sample size will be \(t = \dfrac{\text{mean difference}}{\text{standard deviation}/\sqrt{n}}\). Our sample size n is 24. We will calculate two t-test statistics, one for the standard deviation 10 (for Part a) and one for the standard deviation 25 (for Part b).

Find the P-Value

After finding the test statistic, next step is to find the P-Value. P-Value is the probability of getting a result as extreme (or more) as the observed results under the null hypothesis. We seek to compare P-Value to the given significance level (.05).

Make a Conclusion

Finally, if the P-Value is less than the significance level (.05), we reject the null hypothesis in favor of the alternative one. If it’s more, we fail to reject the null hypothesis. That would provide answers for parts a & b respectively.

Explain Difference in Conclusions

For part c, we need to provide reasons why the conclusions might differ between parts a and b. This will involve an explanation of how the standard deviation affects the outcome of a hypothesis test.

Key concepts

Null Hypothesis

The null hypothesis is the default assumption in a hypothesis test that there is no effect or no difference. In statistics, it's a statement that the parameter in the population—like the mean or the difference between two group means—is equal to a certain value, usually zero. For example, when comparing two mouse designs, the null hypothesis posits that there is no difference in wrist extension between using mouse type A and mouse type B. Formally, this can be expressed as H0: \(\mu = 0\). The purpose of the null hypothesis is to provide a baseline that we can test against using data.

Alternative Hypothesis

The alternative hypothesis, denoted as Ha, is what you want to prove to be true if the null hypothesis is rejected. It's a statement that indicates there is an actual effect or a difference. In our mouse example, the alternative hypothesis suggests that mouse type A results in greater wrist extension than mouse type B, and is formalized as Ha: \(\mu > 0\). When you perform a hypothesis test, you're essentially checking if there's enough evidence to support the alternative hypothesis.

t-test Statistics

The t-test statistic is a ratio calculated from the data that measures the degree of difference between the sample mean and the null hypothesis value, relative to the sample variability. In simpler terms, it shows how many standard errors the data is away from the null hypothesis mean. The formula for the t-test in the mouse study is t = \(\frac{\text{mean difference}}{\text{standard deviation}/\sqrt{n}}\), where n is the sample size. The larger the absolute value of the t-test statistic, the more evidence against the null hypothesis.

P-Value

The P-Value is the probability of obtaining the observed results, or more extreme results, if the null hypothesis is true. A low P-Value, typically less than a set significance level, suggests that such an extreme result is unlikely under the null hypothesis. This leads us to believe that the alternative hypothesis might be true. In essence, the P-Value measures how surprising the data is, given the null hypothesis, and helps us make a decision about which hypothesis to support.

Significance Level

The significance level, often represented by \(\alpha\), is a threshold used to determine whether to reject the null hypothesis. It's the maximum probability of wrongly rejecting the null hypothesis, known as a Type I error. Commonly used significance levels include 0.05, 0.01, and 0.10. In the mouse example, a significance level of 0.05 means there's a 5% chance of concluding that there is a difference in wrist extension between the mouse types when there is no actual difference.