91Ó°ÊÓ

Two proposed computer mouse designs were compared by recording wrist extension in degrees for 24 people who each used both mouse designs ("Comparative Study of Two Computer Mouse Designs," Cornell Human Factors Laboratory Technical Report RP7992). The difference in wrist extension was calculated by subtracting extension for mouse type \(\mathrm{B}\) from the wrist extension for mouse type \(\mathrm{A}\) for each person. The mean difference was reported to be 8.82 degrees. Assume that this sample of 24 people is 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 \(\mathrm{B}\) ? Use a 0.05 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 \(\mathrm{B}\) ? Use a 0.05 significance level. c. Briefly explain why different conclusions were reached in the hypothesis tests of Parts (a) and (b).

Step by step solution

01

Identify the null and alternative hypotheses

For this exercise, the null hypothesis \(H_0\) would state that there is no difference in the mean wrist extension for mouse type A and B. This can mathematically be expressed as \( \mu_{A} - \mu_{B} = 0 \). The alternative hypothesis \(H_1\) would state that the mean extension for mouse type A is greater than mouse type B. This can mathematically be expressed as \( \mu_{A} - \mu_{B} > 0 \). The significance level is 0.05 (or 5%), meaning there is a 5% chance of rejecting the null hypothesis when it's true.

02

Calculate the test statistic and P-value (Standard deviation 10 degrees)

First the z statistic is calculated using the formula z = (sample mean - population mean) / (standard deviation/√n). Here, the sample mean is 8.82 degrees, population mean is 0 (from null hypothesis), standard deviation is 10 degrees and n=24 (number of samples). Plugging these values, we get z ~ 1.76. Then, the P-value is looked from z table which gives us the probability that z scores will be less than our calculated z value. P-value ~ 0.039.

03

Make decision based on the P-value (Standard deviation 10 degrees)

Since the P-value is less than the significance level (0.039 < 0.05), the null hypothesis is rejected. There is enough evidence to support that the mean wrist extension for mouse type A is greater than mouse type B.

04

Calculate the test statistic and P-value (Standard deviation 26 degrees)

Now we will repeat step 2 but with the new standard deviation of 26 degrees. Again calculating, we get z ~ 0.67. From the z table, P-value ~ 0.252.

05

Make decision based on the P-value (Standard deviation 26 degrees)

Since this time the P-value is greater than the significance level (0.252 > 0.05), we cannot reject the null hypothesis. With this standard deviation, there's not enough evidence to support that the mean wrist extension for mouse type A is greater than mouse type B.

06

Analyze different conclusions in parts (a) and (b)

The reason for different conclusions in these two parts is due to the change in standard deviation. A larger standard deviation implies more variability in data which makes it harder to reject the null hypothesis. In part (a), with a smaller standard deviation, the differences are more concentrated making it easier to reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Significance

Statistical significance is a key concept in hypothesis testing. It helps us determine if the observed effect or difference is due to chance or if it is likely to be a true effect. In the context of our exercise, we set a threshold, known as the significance level, often denoted as \(\alpha\), which was 0.05 in this case, meaning there is a 5% risk of concluding that a difference exists when there is no actual difference.
For the mouse design study, we calculate a test statistic and compare the resulting probability, called a P-value, against our significance level. If the P-value is less than \(0.05\), it indicates that the observed difference in mean wrist extensions is statistically significant, suggesting that mouse type A indeed results in greater wrist extension than mouse type B, beyond what would be expected by random chance.
Therefore, statistical significance acts as a filter to help us decide what changes in data should be considered meaningful, validating our hypothesis results.

Confidence Interval

In hypothesis testing, confidence intervals provide a range of values within which we believe the true population parameter lies, with a certain level of confidence. For instance, a 95% confidence interval implies that if the experiment were repeated 100 times, the true mean difference in wrist extension would fall within this range 95 times out of 100. This gives us a sense of the estimate's reliability.
While the exercise focuses on hypothesis testing, understanding confidence intervals is vital. They give context to the mean difference of 8.82 degrees calculated for the mouse designs.
However, the specific confidence intervals weren't calculated in this exercise. But if they were, narrower intervals would imply more precise estimates, leading to stronger confidence in conclusions about mouse type A's wrist extension impact. Hence, confidence intervals complement the hypothesis test by showing the precision of our estimates.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells us how much the results differ from the average value. In this mouse design study, standard deviation plays a crucial role because it affects the test statistic and hence our conclusion about the significance of results.
For example, with a standard deviation of 10 degrees, we found a significant difference in wrist extensions between the two mouse types, rejecting the null hypothesis. However, when the standard deviation increased to 26 degrees, the results were inconclusive, as the test statistic decreased, leading to a higher P-value.
This demonstrates that larger standard deviations create more spread in the data, making it harder to confidently assert a difference in means. Thus, controlling and understanding standard deviation is essential in experiments, as it influences the reliability and outcome of statistical tests.