Question

Manual Dexterity Differences 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

There is no significant difference in the manual dexterity mean scores at \(\alpha = 0.01\).

Step-by-step solution

State the Hypotheses

We begin by stating the null hypothesis and the alternative hypothesis:- Null Hypothesis \(H_0\): There is no difference in the manual dexterity mean scores between athletes and band members, \( \mu_1 = \mu_2 \).- Alternative Hypothesis \(H_a\): There is a difference in the manual dexterity mean scores, \( \mu_1 eq \mu_2 \).

Identify the Test Statistic

Since the population standard deviation is known and the sample sizes are both 30, we use a Z-test to evaluate the difference in two 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 \(n_1 = n_2 = 30\).

Calculate the Test Statistic

Substitute the values into the formula:\[ Z = \frac{(87 - 92)}{\sqrt{\frac{7.2^2}{30} + \frac{7.2^2}{30}}} = \frac{-5}{\sqrt{\frac{51.84}{30} + \frac{51.84}{30}}} = \frac{-5}{\sqrt{3.456 + 3.456}} \]\[ Z = \frac{-5}{\sqrt{6.912}} = \frac{-5}{2.628} \approx -1.902 \]

Determine the Critical Value

Using a significance level \(\alpha = 0.01\) for a two-tailed test, the critical Z-values are \(-2.576\) and \(2.576\).

Make a Decision

The calculated Z-value is approximately \(-1.902\), which lies between \(-2.576\) and \(2.576\). Since the Z-value does not fall in the rejection region, we fail to reject the null hypothesis.

Conclusion

At \(\alpha = 0.01\), there is not enough evidence to suggest a significant difference in the manual dexterity mean scores between the athletes and band members.

Key concepts

Manual Dexterity

Manual dexterity refers to the skill and ease with which someone can perform tasks with their hands. This could range from intricate tasks like playing a musical instrument to practical jobs like typing or repairing small objects.
Dexterity is crucial in many activities, and proficiency can vary significantly between individuals or groups based on training and experience. For instance, athletes may develop specific hand-eye coordination that enhances their manual dexterity in sports. Meanwhile, band members might hone their dexterity through musical training, which emphasizes precision and speed.
Understanding these differences can be key for researchers focusing on how training in diverse fields might influence overall dexterity skills.

Z-test

The Z-test is a statistical method used to determine whether there is a significant difference between the means of two groups. It is particularly useful when the population standard deviations are known and the sample sizes are large (usually over 30).
In the context of hypothesis testing, the formula for the Z-test ensures the calculation of a Z-score, which indicates how many standard deviations the observed difference is from the expected difference under the null hypothesis. This is crucial when assessing the potential differences between groups, such as athletes and band members in this exercise.
The formula used is:

• \[ Z = \frac{(\bar{X}_1 - \bar{X}_2)}{\sqrt{\frac{\sigma^2}{n_1} + \frac{\sigma^2}{n_2}}} \]

By comparing the Z-score to critical values from Z-distribution tables, researchers can judge whether observed differences are statistically significant.

Null Hypothesis

The null hypothesis is a foundational concept in hypothesis testing. It proposes that there is no effect or difference in the situation being analyzed.
For instance, if we want to compare whether athletes and band members have different manual dexterity scores, the null hypothesis asserts that any observed difference in their average scores is due to random chance or sampling error.
In mathematical terms, for this exercise, the null hypothesis is represented as:

• \( H_0: \mu_1 = \mu_2 \)

Failing to reject the null hypothesis does not prove it true. It simply indicates insufficient evidence to declare a statistical difference at the chosen significance level.

Significance Level

The significance level, often denoted by \( \alpha \), is a threshold used in hypothesis testing that determines when to reject the null hypothesis. It quantifies the risk you are willing to take of incorrectly rejecting a true null hypothesis.
A common choice for \( \alpha \) is 0.05, but in more stringent tests, researchers might select 0.01, as seen in this exercise.
A significance level of 0.01 implies that there is a 1% risk of concluding that there is a difference when there isn't one. In a two-tailed test, like this one, critical values for \( \alpha = 0.01 \) are determined, and the Z-score is compared to these values to decide on the hypothesis. If the Z-score falls outside these range values, the null hypothesis is rejected, indicating a statistically significant difference.