Question

Perform each of the following steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. A researcher wishes to see if there is a difference in the manual dexterity of athletes and that of band members. Two random samples of 30 are selected from each group and are given a manual dexterity test. The mean of the athletes' test was \(87,\) and the mean of the band members' test was \(92 .\) The population standard deviation for the test is \(7.2 .\) At \(\alpha=0.01,\) is there a significant difference in the mean scores?

Short answer

At \(\alpha = 0.01\), there is no significant difference in the mean dexterity scores of athletes and band members.

Step-by-step solution

State the Hypotheses

In hypothesis testing, we start with a null hypothesis (H0) and an alternative hypothesis (H1). The null hypothesis (H0) usually states that there is no effect or no difference. In this case:\(H_0: \mu_1 = \mu_2\) (The mean dexterity scores of athletes and band members are equal.) \(H_1: \mu_1 eq \mu_2\) (The mean dexterity scores of athletes and band members are different.)The alternative hypothesis (H1) suggests there is a difference, hence this is a two-tailed test.

Identify the Claim

The claim in this statement is from the alternative hypothesis, which asserts that there is a difference in the dexterity scores of athletes and band members (\(H_1: \mu_1 eq \mu_2\)).

Find the Critical Value(s)

For a two-tailed test with a significance level \(\alpha = 0.01\), we need to find the critical z-values from the standard normal distribution table. Since \(\alpha = 0.01\), we have \(\alpha/2 = 0.005\) in each tail.The critical z-values corresponding to \(\alpha/2 = 0.005\) are approximately \(z = \pm 2.576\). Thus, the critical region for rejection of the null hypothesis is \(z < -2.576\) or \(z > 2.576\).

Compute the Test Value

To compute the test statistic value, we use the formula for the z-test for two sample means:\[z = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{\sigma^2}{n_1} + \frac{\sigma^2}{n_2}}}\]Where \(\bar{x}_1 = 87\), \(\bar{x}_2 = 92\), \(\sigma = 7.2\), and the sample sizes are \(n_1 = 30\) and \(n_2 = 30\).\[z = \frac{(87 - 92)}{\sqrt{\frac{7.2^2}{30} + \frac{7.2^2}{30}}} = \frac{-5}{\sqrt{\frac{51.84}{30} + \frac{51.84}{30}}} = \frac{-5}{\sqrt{3.456 + 3.456}} = \frac{-5}{\sqrt{6.912}} \approx \frac{-5}{2.629} \approx -1.902\]The test statistic value is \(z \approx -1.902\).

Make the Decision

We compare the test statistic value \(z = -1.902\) to the critical values \(-2.576\) and \(2.576\). Since \(-1.902\) is not less than \(-2.576\) and not greater than \(2.576\), it does not fall into the rejection region. Thus, we do not reject the null hypothesis \(H_0\).

Summarize the Results

Since we did not reject the null hypothesis, we conclude that there is not sufficient evidence at \(\alpha = 0.01\) to support the claim that there is a significant difference in the mean manual dexterity scores of athletes and band members.

Key concepts

Null Hypothesis

In hypothesis testing, one of the first steps is to formulate the null hypothesis. This hypothesis, symbolized as \(H_0\), generally posits that there is no effect or difference between groups or variables being tested. It acts as the baseline assumption, suggesting that any observed difference is due to random variation or chance rather than a true effect.

The null hypothesis is crucial because it provides a statement that can be tested statistically. In the exercise, the null hypothesis \(H_0: \mu_1 = \mu_2\) asserts that the mean manual dexterity scores of athletes and band members are equal. By conducting hypothesis testing, researchers aim to find evidence against \(H_0\), thereby providing support for the alternative hypothesis \(H_1\).

Ultimately, the null hypothesis serves as the starting point in statistical tests and helps guide the decision-making process by clearly defining what "no difference" looks like.

Critical Value

A critical value in hypothesis testing is a threshold that determines the boundary of the rejection region. It helps decide whether to reject the null hypothesis \(H_0\). Critical values are derived from the significance level \(\alpha\), which reflects the probability of rejecting \(H_0\) when it is true, known as a Type I error.

In a two-tailed test, the significance level \(\alpha\) is split between two tails of the distribution. For the exercise, with a significance level \(\alpha = 0.01\), each tail has an \(\alpha/2 = 0.005\). The critical values are the z-scores that correspond to the tails of this standard normal distribution.

The critical values for the exercise are approximately \(z = \pm 2.576\). These values define the bounds of the rejection region, where \(|z| > 2.576\) would lead to rejecting the null hypothesis. Understanding critical values is significant because they provide a clear criterion to judge the test statistic against and make a definitive decision.

Test Statistic

The test statistic is a standardized value calculated from sample data during a hypothesis test. It measures how far the sample data diverge from what is expected under the null hypothesis \(H_0\). In our exercise, the test statistic is calculated using the z-test formula for two sample means, as we deal with a known population standard deviation.

The formula given is:

• \(z = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{\sigma^2}{n_1} + \frac{\sigma^2}{n_2}}}\)

Where \(\bar{x}_1 = 87\), \(\bar{x}_2 = 92\), \(\sigma = 7.2\), and \(n_1 = n_2 = 30\). After substitution and simplification:

• \(z \approx -1.902\)

The calculated test statistic \(z\) is then compared to critical values to infer whether the null hypothesis should be rejected. The test statistic essentially quantifies the degree of deviation between the sample observations and the null hypothesis.

Two-Tailed Test

A two-tailed test refers to a scenario in hypothesis testing where the researcher is interested in detecting differences in both directions – whether the parameter of interest is greater than or less than a certain value. Here, the parameter of interest is the mean manual dexterity scores of athletes and band members.

The alternative hypothesis \(H_1: \mu_1 eq \mu_2\) implies the possibility of either increase or decrease in one mean compared to the other. This makes it a two-tailed test, as we are checking for deviations in both directions.

In such tests, the significance level \(\alpha\) is divided equally into two, with \(\alpha/2\) in each tail of the distribution. This determines the critical region's lower and upper bounds, beyond which \(H_0\) will be rejected. Two-tailed tests are comprehensive and allow for checking more flexibility regarding the differences.