Question

Do Hands Adapt to Water? Researchers in the UK designed a study to determine if skin wrinkled from submersion in water performed better at handling wet objects. \(^{62}\) They gathered 20 participants and had each move a set of wet objects and a set of dry objects before and after submerging their hands in water for 30 minutes (order of trials was randomized). The response is the time (seconds) it took to move the specific set of objects with wrinkled hands minus the time with unwrinkled hands. The mean difference for moving dry objects was 0.85 seconds with a standard deviation of 11.5 seconds. The mean difference for moving wet objects was -15.1 seconds with a standard deviation of 13.4 seconds. (a) Perform the appropriate test to determine if the wrinkled hands were significantly faster than unwrinkled hands at moving dry objects. (b) Perform the appropriate test to determine if the wrinkled hands were significantly faster than unwrinkled hands at moving wet objects.

Short answer

For the dry objects, if the calculated t is greater than the critical value, reject the null hypothesis which means the wrinkled hands are significantly faster. For the wet objects, if the calculated t is less than critical value, reject the null hypothesis which means the wrinkled hands are significantly faster.

Step-by-step solution

Define the Hypotheses for Part (a)

For the first part, the null hypothesis \(H_{0}: \mu_{D} = 0\), where \(\mu_{D}\) is the mean difference for moving dry objects. The alternative hypothesis \(H_{1}: \mu_{D} > 0\).

Compute the Test Statistic for Part (a)

For the test statistic, calculate \(t = \frac{\bar{X}-\mu_{0}}{s / \sqrt{n}}\), where \(\bar{X}\) is the sample mean (0.85), \(\mu_{0}\) is the null hypothesis mean (0), s is the standard deviation (11.5), and n is the sample size (20).

Determine the Critical Value and Make a Decision for Part (a)

The critical value determined from t-distribution table with \(df = n - 1\) (19) at the 5% level of significance for a one-tailed test. If computed t is greater than the critical value, reject the null hypothesis.

Define the Hypotheses for Part (b)

For the second part, the null hypothesis \(H_{0}: \mu_{W} = 0\), where \(\mu_{W}\) is the mean difference for moving wet objects. The alternative hypothesis \(H_{1}: \mu_{W} < 0\).

Compute the Test Statistic for Part (b)

The test statistic is calculated in the same way but using the sample mean (-15.1), the standard deviation (13.4), and the sample size (20).

Determine the Critical Value and Make a Decision for Part (b)

The critical value is determined in the same way as in step 3 but the t-value is compared for the left tail of the distribution. If the calculated t is less than the critical value, reject the null hypothesis.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is fundamental to hypothesis testing. In the context of our exercise, hypothesis testing seeks to determine whether the observed effects of wrinkled hands on moving objects are due to chance or a real effect. The null hypothesis (\( H_0 \)) represents a statement of no effect or no difference, essentially a baseline or default assumption. In part (a) of our exercise, the null hypothesis is that the mean difference in time for moving dry objects with wrinkled versus unwrinkled hands is zero (\( \mu_D = 0 \) ).

The alternative hypothesis (\( H_1 \)), on the other hand, represents what the researchers really want to test; it is the statement of an effect or difference. In the case of moving dry objects, the alternative hypothesis is that the mean difference is greater than zero (\( \mu_D > 0 \) ), indicating that wrinkled hands were faster. Similarly, for part (b) the alternative hypothesis postulates that the mean difference in time for moving wet objects with wrinkled hands is less than zero (\( \mu_W < 0 \) ), suggesting that wrinkled hands could be significantly faster at this task.

These hypotheses set the stage for the entire test, guiding the calculations and interpretations that follow.

Test Statistic Calculation

The test statistic is pivotal to hypothesis testing as it helps us decide whether to reject the null hypothesis. It is a standardized value that measures the degree of deviation from the null hypothesis. To compute the test statistic in part (a) for our exercise, we use the formula \( t = \frac{\bar{X}-\mu_{0}}{s / \sqrt{n}} \) where \(\bar{X}\) is the sample mean, \(\mu_{0}\) is the hypothesized population mean under the null, \(s\) is the sample standard deviation, and \(n\) is the sample size.

For part (a), \(\bar{X}\) is 0.85 seconds. The test statistic measures how many standard errors the sample mean is away from the null hypothesis mean. It considers both the size of the effect (the sample mean difference) and the precision of the estimate (the sample size and standard deviation). Plugging in the values, we get a specific test statistic that we can compare against a critical value to make a statistical decision.

t-Distribution Critical Value

The t-distribution critical value acts as a threshold in hypothesis testing. To determine whether our test statistic is extreme enough to reject the null hypothesis, we must compare it to a critical value from the t-distribution. This critical value depends on the desired level of significance (commonly 5%) and the degrees of freedom, which in turn depend on the sample size.

In the exercise, the degrees of freedom (df) for each test is 19, since it's calculated as the sample size minus one (\( n - 1 \)). Once we know the df, we can look up the critical value in a t-distribution table or use statistical software for more precise results. At a 5% significance level, the critical value is the point in the t-distribution where there's only a 5% chance of observing a more extreme value if the null hypothesis is true. The decision rule is straightforward: if the test statistic exceeds the critical value in a one-tailed test, we reject the null hypothesis.

One-tailed Test Significance

Finally, the concept of one-tailed test significance refers to the direction of the hypothesis. In a one-tailed test, the alternative hypothesis states that the parameter of interest is either strictly greater than or less than the null hypothesis value. It implies that researchers have a specific direction in mind for the effect being tested.

In our exercise, part (a) is a right-tailed test because we are testing if wrinkled hands are faster (a positive effect) at moving dry objects, so we are looking for a test statistic that exceeds the critical value on the right side of the t-distribution. Conversely, part (b) is a left-tailed test where we check if wrinkled hands are faster at moving wet objects, which corresponds to a negative mean difference (a negative effect), and we seek a test statistic that falls below the critical value on the left side. One-tailed tests are powerful when a specific direction is hypothesized because they can detect an effect in that direction with less data, increasing the test’s sensitivity.