Question

10.52 - Medical research has shown that repeated wrist extension beyond 20 degrees increases the risk of wrist and hand injuries. Each of 24 students at Cornell University used a proposed new computer mouse design, and while using the mouse, each student's wrist extension was recorded. Data consistent with summary values given in the paper "Comparative Study of Two Computer Mouse Designs" (Cornell Human Factors Laboratory Technical Report \(\mathrm{RP} 7992\) ) are given. Use these data to test the hypothesis that the mean wrist extension for people using this new mouse design is greater than 20 degrees. Are any assumptions required in order for it to be appropriate to generalize the results of your test to the population of Cornell students? To the population of all university students? \(\begin{array}{llllllllllll}27 & 28 & 24 & 26 & 27 & 25 & 25 & 24 & 24 & 24 & 25 & 28 \\ 22 & 25 & 24 & 28 & 27 & 26 & 31 & 25 & 28 & 27 & 27 & 25\end{array}\)

Short answer

After doing the computations, if the p-value is less than the chosen significance level, the null hypothesis is rejected and hence, the mean wrist extension with this new mouse design is found to be significantly greater than 20 degrees. The assumption of independency and sample representation must hold for results to be generalized to the Cornell or all-university student populations.

Step-by-step solution

Define Hypotheses

First define the null hypothesis \(H_0\) that the mean wrist extension is 20 degrees: \(H_0: \mu = 20\), and the alternative hypothesis \(H_a\) that it is more than 20 degrees: \(H_a: \mu > 20\).

Compute Sample Mean

The sample mean is the sum of all wrist extensions divided by the number of observations. Use this formula: \(\overline{X} = \frac{1}{n}\sum_{i=1}^{n} x_i\). Calculate it with the given data values.

Calculate Sample Standard Deviation

The sample standard deviation is calculated using the formula \(s = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}\). Do this calculation using the data from the observations.

Calculate Test Statistic

To calculate the test statistic, use the sample mean, population mean under null hypothesis, sample standard deviation and sample size in this formula: \(t = \frac{\overline{X} - 20}{s/\sqrt{n}}\). The test statistic follows approximately a t-distribution with n-1 degrees of freedom.

Find P-value

Use a t-table or statistical software to find the p-value, which is the probability of observing a test statistic as extreme as the one calculated (or more extreme), assuming the null hypothesis is true. Because this is a right-tailed test, we find the area to the right of our calculated t-value.

Conclusion

If the p-value is less than the chosen significance level (typically 0.05), reject the null hypothesis and conclude that the mean wrist extension is significantly greater than 20 degrees.

Assumptions and Generalizability

Assumptions include independence of observations and the sample being representative of the relevant student population (e.g., Cornell or all universities). If these hold, it's appropriate to generalize results to these populations.

Key concepts

Null Hypothesis

In the realm of hypothesis testing, the **Null Hypothesis** is a pivotal concept. It acts as the anchor of our testing procedure, representing a statement that we assume to be true until evidence suggests otherwise. Specifically, for the wrist extension study, the null hypothesis (\( H_0 \)) posits that the mean wrist extension angle for users of the new mouse design is exactly 20 degrees. More generally:

• The null hypothesis asserts there is no effect or no difference (i.e., no deviation from a standard value, such as 20 degrees in this case).
• We represent it using the symbol \( \mu \), which stands for the population mean.
• In mathematical terms, it is expressed as \( H_0: \mu = 20 \) degrees.

In hypothesis tests, the null hypothesis is our baseline. Any deviation in our observations against this hypothesis indicates whether we have enough evidence to support an alternative hypothesis.

T-Distribution

The **T-Distribution** is a fundamental concept when conducting hypothesis tests on sample data, especially small sample sizes, like in our wrist extension example. It helps determine how data points are spread out in relation to the mean. Here’s how it fits into the testing process:

• The t-distribution accounts for the variability that arises when sampling from a normal distribution.
• This distribution is particularly useful when the sample size is small (typically less than 30).
• It resembles a normal distribution but has thicker tails, allowing for more data variation.

In this study:

• The test statistic calculated from the sample data follows an approximate t-distribution with degrees of freedom (df) equal to the sample size minus one.
• For our 24 students, df would be 23.

The t-distribution helps us determine critical values to assess the null hypothesis. We rely on it to compute the probability of observing a test statistic at least as extreme as ours, assuming the null hypothesis holds true.

Sample Mean

The **Sample Mean** is an essential statistic that serves as an estimate of the population mean. It is the average of all measured observations in a sample. For the wrist extension task:

• The sample mean was computed by adding all wrist extension measurements and dividing by the total number of students.
• The formula used is: \( \overline{X} = \frac{1}{n}\sum_{i=1}^{n} x_i \), where \( \overline{X} \) is the sample mean, and \( x_i \) represents each individual measurement.

Understanding the sample mean helps gauge central tendency within the data. Comparing it against the hypothesized population mean (20 degrees) is critical in hypothesis testing. A substantial difference suggests the sample mean does not reflect what is expected under the null hypothesis.

Sample Standard Deviation

The **Sample Standard Deviation** measures the extent of variability or dispersion of individual observations around the sample mean. It is a key metric in hypothesis testing that allows us to quantify uncertainty:

• Calculated using the formula: \( s = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}} \), where \( x_i \) are the sample observations, and \( \bar{x} \) is the sample mean.
• This value gives insight into how spread out the wrist extension measurements are from the sample mean.

The sample standard deviation plays a crucial role in computing the test statistic for hypothesis testing. It acts as the denominator in the t-statistic formula, providing scale to measure how significant the difference between the sample mean and the hypothesized population mean is. This ultimately helps in deciding whether to accept or reject the null hypothesis.