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Chapter 10: Problem 53

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 mouse design, and while using the mouse their wrist extension was recorded for each one. Data consistent with summary values given in the paper "Comparative Study of Two Computer Mouse Designs" (Cornell Human Factors Laboratory Technical Report RP7992) 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? (data on next page) $$ \begin{array}{llllllllllll} 27 & 28 & 24 & 26 & 27 & 25 & 25 & 24 & 24 & 24 & 25 & 28 \\ 22 & 25 & 24 & 28 & 27 & 26 & 31 & 25 & 28 & 27 & 27 & 25 \end{array} $$

Short Answer

Expert verified
The conclusion on rejecting or failing to reject the null hypothesis will depend on the calculated test statistic and the determined critical value. The results can be generalized to the population of all Cornell students or all university students only if certain assumptions like representativeness of the sample and normality of the data are met.

Step by step solution

01

Calculation of Sample Mean and Standard Deviation

From the given data, calculate the sample mean \(\bar{x}\) and the sample standard deviation \(s\). Use the formulas \(\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_{i}\) for the mean and \(s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\bar{x})^2}\) for the standard deviation, where \(n\) is the number of observations, and \(x_{i}\) are the individual observations.
02

Set Up Hypotheses

The null and alternative hypotheses are set up as follows: Null hypothesis \(H0: \mu \leq 20\) Alternative hypothesis \(H1: \mu > 20\) Here, \(\mu\) is the population mean.
03

Calculate the Test Statistic

Calculate the test statistic (t) using the formula: \(t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\) Here, \(\mu_0\) is the value of the population mean under the null hypothesis, which is 20 in this case.
04

Determine the Critical Value and Make a Decision

Using a standard t-table or a calculator, find the t critical value for the given level of significance and degree of freedom (\(df = n-1\)). If the calculated test statistic is greater than the critical value, you reject the null hypothesis. Otherwise, you fail to reject the null hypothesis.
05

Discuss the Assumptions

Discuss the assumptions of the model. To generalize the results to a larger population, it is assumed that the sample is representative of the population and that the data follows a normal distribution.

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Key Concepts

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

Sample Mean
When you want to understand the average of a set of numbers, you calculate the sample mean. It tells us the central tendency, or the common value among the data points.
To find the sample mean, use the formula:
  • Sum up all the values in the data set.
  • Divide the total by the number of values (n).
This provides an overall picture of your data's behavior. For our exercise, the mean wrist extension angle is what we're interested in, to see if it surpasses the 20-degree mark.
Standard Deviation
Standard deviation measures how spread out the numbers in a data set are. If data points are close to each other, the standard deviation is low; if they are scattered, it's high.
It's calculated by finding the square root of the average squared difference from the mean. The formula is:
  • Subtract the mean from each data point and square the result.
  • Find the mean of these squared differences.
  • Take the square root of this average.
In our problem, we compute the standard deviation to understand the variability of wrist extension angles among students.
Test Statistic
A test statistic assesses whether your sample data supports a hypothesis. In our context, it's used to determine if the mean wrist extension is greater than 20 degrees.
The formula for the t-test statistic is:
  • Subtract the hypothesized mean (20 degrees) from the sample mean.
  • Divide the result by the standard deviation divided by the square root of the sample size (n).
This calculation helps compare your observed sample mean to the hypothesized population mean.
Null and Alternative Hypothesis
Hypothesis testing involves constructing two hypotheses to compare. The null hypothesis ( H_0 ) generally represents the default or status quo, and the alternative hypothesis ( H_1 ) represents the claim we want to test.
In our example, the hypotheses are:
  • Null ( H_0 ): The mean wrist extension is 20 degrees or less.
  • Alternative ( H_1 ): The mean wrist extension is greater than 20 degrees.
These hypotheses guide the direction of our analysis, helping us decide if there is significant evidence to support the claim that the new mouse design increases wrist extension.

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Most popular questions from this chapter

In a study of computer use, 1000 randomly selected Canadian Internet users were asked how much time they spend using the Internet in a typical week (Ipsos Reid, August 9,2005 ). The mean of the 1000 resulting observations was \(12.7\) hours. a. The sample standard deviation was not reported, but suppose that it was 5 hours. Carry out a hypothesis test with a significance level of \(.05\) to decide if there is convincing evidence that the mean time spent using the Internet by Canadians is greater than \(12.5\) hours. b. Now suppose that the sample standard deviation was 2 hours. Carry out a hypothesis test with a significance level of \(.05\) to decide if there is convincing evidence that the mean time spent using the Internet by Canadians is greater than \(12.5\) hours. c. Explain why the null hypothesis was rejected in the test of Part (b) but not in the test of Part (a).

Newly purchased automobile tires of a certain type are supposed to be filled to a pressure of 30 psi. Let \(\mu\) denote the true average pressure. Find the \(P\) -value associated with each of the following given \(z\) statistic values for testing \(H_{0}: \mu=30\) versus \(H_{a}: \mu \neq 30\) when \(\sigma\) is known: a. \(2.10\) d. \(1.44\) b. \(-1.75\) e. \(-5.00\) c. \(0.58\)

Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most \(150^{\circ} \mathrm{F}\), there will be no negative effects on the river's ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above \(150^{\circ} \mathrm{F}\), researchers will take 50 water samples at randomly selected times and record the temperature of each sample. The resulting data will be used to test the hypotheses \(H_{0}: \mu=150^{\circ} \mathrm{F}\) versus \(H_{a}: \mu>150^{\circ} \mathrm{F}\). In the context of this example, describe Type I and Type II errors. Which type of error would you consider more serious? Explain.

An automobile manufacturer who wishes to advertise that one of its models achieves \(30 \mathrm{mpg}\) (miles per gallon) decides to carry out a fuel efficiency test. Six nonprofessional drivers are selected, and each one drives a car from Phoenix to Los Angeles. The resulting fuel efficiencies (in miles per gallon) are: \(\begin{array}{llllll}27.2 & 29.3 & 31.2 & 28.4 & 30.3 & 29.6\end{array}\) Assuming that fuel efficiency is normally distributed under these circumstances, do the data contradict the claim that true average fuel efficiency is (at least) \(30 \mathrm{mpg}\) ?

Past experience has indicated that the true response rate is \(40 \%\) when individuals are approached with a request to fill out and return a particular questionnaire in a stamped and addressed envelope. An investigator believes that if the person distributing the questionnaire is stigmatized in some obvious way, potential respondents would feel sorry for the distributor and thus tend to respond at a rate higher than \(40 \%\). To investigate this theory, a distributor is fitted with an eye patch. Of the 200 questionnaires distributed by this individual, 109 were returned. Does this strongly suggest that the response rate in this situation exceeds the rate in the past? State and test the appropriate hypotheses at significance level \(.05\).
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