Question

Two proposed computer mouse designs were compared by 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 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 \(A\) is greater than for mouse type B? Use a .05 significance level. b. Suppose that the standard deviation of the differences was 26 degrees. Is there convincing evidence that the mean wrist extension for mouse type \(A\) 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

Without the actual calculations, an answer cannot be provided. However, it's possible that for a standard deviation of 10 degrees, there may be convincing evidence that the mean wrist extension for mouse \(type A\) is greater than for mouse type \(B\) while for a standard deviation of 26 degrees, the evidence may not be as convincing due to higher variability in the data. The computed t-scores and corresponding p-values for both cases will provide a clear answer.

Step-by-step solution

Define the Hypotheses

The first step in a hypothesis test is to define the null hypothesis and the alternative hypothesis. Here, for both cases: \n\n-The null hypothesis (\(H_0\)) is that the mean wrist extension for mouse type \(A\) is NOT greater than for mouse type \(B\), which can be mathematically represented as: \(\mu_A - \mu_B \leq 0\) \n\n-The alternative hypothesis (\(H_A\)) is that the mean wrist extension for mouse type \(A\) IS greater than for mouse type \(B\), represented as: \(\mu_A - \mu_B > 0\)

Calculate the Test Statistic (part a)

For case a, with a standard deviation of 10, calculate the test statistic using the formula for the t-score: \(t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}\). Here, \(\bar{x}\) is the sample mean (8.82), \( \mu_0\) is the value under the null hypothesis (0), \(s\) is the standard deviation (10), and \(n\) is the sample size (24). Calculate the t-score and then use a t-distribution table (or an online calculator) to find the p-value associated with this t-test statistic.

Make a Decision (part a)

If the calculated p-value is less than the significance level (0.05), then reject the null hypothesis in favor of the alternative. This would indicate there is convincing evidence that the mean wrist extension for mouse type \(A\) is greater than for mouse type \(B\).

Calculate the Test Statistic (part b)

Perform the same calculation as in Step 2, but now with the standard deviation being 26.

Make a Decision (part b)

Again, if the calculated p-value is less than the significance level (0.05), then reject the null hypothesis in favor of the alternative. If the p-value is high, failing to reject the null hypothesis would indicate that there isn't strong evidence that mouse type \(A\) leads to greater wrist extension than mouse type \(B\).

Discuss the Conclusion Difference

Explain why different conclusions were reached in the hypothesis tests of Parts (a) and (b). A key point is the effect of larger standard deviation on the t-test statistic and subsequently on the p-value. When the standard deviation is large, even if the mean difference is substantial, it may not be statistically significant due to the increased variability of the data. The smaller the standard deviation, the smaller our denominator in the t-score formula, which could result in a larger t-value and stronger evidence against the null hypothesis.

Key concepts

Null Hypothesis

In hypothesis testing, the null hypothesis (\(H_0\)) is a statement that there is no effect or no difference. It serves as the default or baseline claim that we test against. In our exercise involving computer mouse designs, the null hypothesis posited that there is no significant difference in wrist extension between mouse type A and mouse type B.

• Mathematically, it's often presented as an equal or less-than statement: \( \mu_A - \mu_B \leq 0 \), meaning the mean wrist extension for A is not greater than B.
• The primary goal in hypothesis testing is to either reject or fail to reject this null hypothesis based on statistical evidence.

The null hypothesis is crucial because it simplifies the testing process, creating a foundation to either support or challenge with data.

Alternative Hypothesis

The alternative hypothesis (\(H_A\)) suggests there is a meaningful effect or difference, and it stands in opposition to the null hypothesis. In the comparative study of mouse designs, the alternative hypothesis claimed that mouse type A results in a greater mean wrist extension than mouse type B.

• This is expressed mathematically as: \( \mu_A - \mu_B > 0 \), indicating the aim to prove that A exceeds B.
• The alternative hypothesis is directional in this case, specifically pointing to a greater value, which implies a one-tailed test.

If the data provides enough evidence against the null hypothesis, we support the alternative hypothesis. It's important to note that adopting the alternate hypothesis means we've observed a statistically meaningful difference.

Significance Level

The significance level, represented as alpha (\( \alpha \)), is a threshold set before the experiment begins, denoting the probability of rejecting the null hypothesis when it is actually true (a Type I error). A common choice for \( \alpha \) is 0.05.

• In our exercise, a 0.05 significance level indicates a 5% risk of improperly rejecting the null hypothesis.
• Choosing a significance level is a balance between being too lenient (and mistakenly finding effects that do not exist) or too stringent (and missing actual effects).

Setting this level helps ensure that any decisions made are backed by statistically significant findings rather than arbitrary results, enforcing rigorous standards in hypothesis testing.

t-Test

A t-test is a statistical analysis used to evaluate if there are significant differences between the means of two groups (in some cases, a group compared to a known value). The t-test calculates a t-score to determine the probability that the observed data could occur under the null hypothesis.

• In this exercise, the t-test helps analyze the wrist extension differences between the two types of computer mice.
• The formula for the t-score is: \( t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \), where \( \bar{x} \) is the sample mean, \( \mu_0 \) the hypothesized value (commonly 0), \( s \) the standard deviation, and \( n \) the sample size.

In hypothesis testing, the calculated t-score and associated p-value help decide whether to reject the null hypothesis and accept the alternative hypothesis, based on the pre-set significance level.

Standard Deviation

Standard deviation is a measure of data dispersion or variability within a data set. Understanding how spread out the values are gives insight into the reliability of the mean.

• In our exercise, the standard deviation of the differences in wrist extension was examined at two levels: 10 and 26 degrees.
• A smaller standard deviation (10 degrees) suggests consistent data points close to the mean, leading to a higher likelihood of detecting a true difference between group means with the t-test.
• A larger standard deviation (26 degrees) implies more variation, making it harder to distinguish significant differences, as observed in the differing conclusions reached in part (a) versus part (b).

Being aware of the standard deviation is critical, as it has direct implications on the statistical significance of the results and the conclusions drawn from hypothesis tests.