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Chapter 10: Problem 63

The amount of shaft wear after a fixed mileage was determined for each of seven randomly selected internal combustion engines, resulting in a mean of \(0.0372\) inch and a standard deviation of \(0.0125\) inch. a. Assuming that the distribution of shaft wear is normal, test at level \(.05\) the hypotheses \(H_{0}: \mu=.035\) versus \(H_{a}: \mu>.035 .\) b. Using \(\sigma=0.0125, \alpha=.05\), and Appendix Table 5, what is the approximate value of \(\beta\), the probability of a Type II error, when \(\mu=.04\) ?

Short Answer

Expert verified
The detailed solution will provide the correct critical values and thus whether to reject the null hypothesis. It would also compute the potential risk of making Type II error, expressed as \(\beta\).

Step by step solution

01

State the Null and Alternative Hypotheses

The null hypothesis is \(H_{0}: \mu = 0.035\) while the alternative hypothesis is \(H_{a}: \mu > 0.035\). We are suggesting that the mean wear is greater than 0.035 with the alternative hypothesis.
02

Perform the Hypothesis Test

The z-score is calculated using the formula \((\overline{X} - \mu) / (\sigma/ \sqrt{n})\) \n With the given data, the z-score is \((0.0372 - 0.035) / (0.0125/ \sqrt{7})\). This z-score can be compared to the critical value for alpha=0.05 in a one-tail test. If the z-score is greater than the critical value, we reject the null hypothesis.
03

Calculation of Type II Error

A Type II error occurs when we fail to reject the null hypothesis while it is false. We are given that the actual mean is 0.04, hence to find the beta value, we need to calculate the z-score using the formula \((\mu - \mu_{0}) / (\sigma/ \sqrt{n})\) \n Where, \(\mu_{0}\) is the value from null hypothesis and \(\mu\) is the actual value and calculate the corresponding probability from the z-table.
04

Conclusion

Based on the calculated z-scores and resulting p-values, we can decide whether to reject or fail to reject the null hypothesis for part a, and determine the risk of type II error, i.e., the calculated beta for part b.

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Key Concepts

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

Type I Error
A Type I error occurs when we mistakenly reject a true null hypothesis. Think of it as a false alarm. In the context of hypothesis testing, it's like falsely believing something significant has occurred when, in fact, it has not.
This error is denoted by the symbol \( \alpha \), which is also known as the level of significance. For example, in this exercise, the level of significance \( \alpha \) is set at 0.05.
This means there is a 5% risk of claiming that the mean shaft wear is greater than 0.035 inches, even if it's actually not.
  • Type I errors can lead to incorrect conclusions and misinformed decisions.
  • Selecting an appropriate significance level can help control the likelihood of committing a Type I error.
Choosing a smaller \( \alpha \) reduces the chance of a Type I error but it also makes it harder to detect a true significant effect.
Type II Error
A Type II error happens when we fail to reject a false null hypothesis. This means we miss a true effect, acting like nothing happened when, in fact, it did. In this exercise, we calculate the probability of making a Type II error, denoted by \( \beta \).
If the actual mean is 0.04 inches, the task requests us to determine \( \beta \) under this condition.
  • Type II errors lead to missed discoveries or undetected true effects. They can result in failing to take necessary action when needed.
  • Calculation of \( \beta \) helps understand the test's power: the probability of detecting an effect when there is one to detect.
Increasing sample size or selecting a higher significance level can help reduce the probability of Type II errors.
Z-score
A Z-score translates a data point into how many standard deviations it is from the mean. It's a key element in hypothesis testing as it tells us exactly where our sample mean fits within the distribution.
In our hypothesis test, we calculate the Z-score with the formula \((\overline{X} - \mu) / (\sigma/ \sqrt{n})\).
  • The Z-score helps compare our sample outcome against the critical value derived from a Z-table.
  • If the Z-score is higher than the critical value, we reject the null hypothesis assuming our test is right-tailed.
In this exercise, the Z-score computation helps decide whether the observed shaft wear is significantly greater than 0.035 inches.
Normal Distribution
The normal distribution is a bell-shaped curve that describes how data is spread out, with most data clustering around the mean.
This distribution is crucial in hypothesis testing, as many statistical tests assume data follows a normal distribution. In this exercise, shaft wear is assumed to be normally distributed.
  • It allows us to use Z-scores to establish how typical or unusual a sample result is.
  • Normal distribution is characterized by properties like the empirical rule (68-95-99.7 rule) which indicates that data within one, two, and three standard deviations from the mean captures about 68%, 95%, and 99.7% of the data, respectively.
Understanding and assuming normal distribution permits the straightforward calculation of probabilities and critical values in tests like the one in this problem.

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Most popular questions from this chapter

The paper "Playing Active Video Games Increases Energy Expenditure in Children" (Pediatrics [2009]: 534-539) describes an interesting investigation of the possible cardiovascular benefits of active video games. Mean heart rate for healthy boys age 10 to 13 after walking on a treadmill at \(2.6 \mathrm{~km} /\) hour for 6 minutes is 98 beats per minute (bpm). For each of 14 boys, heart rate was measured after 15 minutes of playing Wii Bowling. The resulting sample mean and standard deviation were \(101 \mathrm{bpm}\) and \(15 \mathrm{bpm}\), respectively. For purposes of this exercise, assume that it is reasonable to regard the sample of boys as representative of boys age 10 to 13 and that the distribution of heart rates after 15 minutes of Wii Bowling is approximately normal. a. Does the sample provide convincing evidence that the mean heart rate after 15 minutes of Wii Bowling is different from the known mean heart rate after 6 minutes walking on the treadmill? Carry out a hypothesis test using \(\alpha=.01\). b. The known resting mean heart rate for boys in this age group is \(66 \mathrm{bpm}\). Is there convincing evidence that the mean heart rate after Wii Bowling for 15 minutes is higher than the known mean resting heart rate for boys of this age? Use \(\alpha=.01\).

A study of fast-food intake is described in the paper "What People Buy From Fast-Food Restaurants" (Obesity [2009]: \(1369-1374\) ). Adult customers at three hamburger chains (McDonald's, Burger King, and Wendy's) at lunchtime in New York City were approached as they entered the restaurant and asked to provide their receipt when exiting. The receipts were then used to determine what was purchased and the number of calories consumed was determined. In all, 3857 people participated in the study. The sample mean number of calories consumed was 857 and the sample standard deviation was 677 . a. The sample standard deviation is quite large. What does this tell you about number of calories consumed in a hamburger-chain lunchtime fast-food purchase in New York City? b. Given the values of the sample mean and standard deviation and the fact that the number of calories consumed can't be negative, explain why it is not reasonable to assume that the distribution of calories consumed is normal. c. Based on a recommended daily intake of 2000 calories, the online Healthy Dining Finder (www .healthydiningfinder.com) recommends a target of 750 calories for lunch. Assuming that it is reasonable to regard the sample of 3857 fast-food purchases as representative of all hamburger-chain lunchtime purchases in New York City, carry out a hypothesis test to determine if the sample provides convincing evidence that the mean number of calories in a New York City hamburger-chain lunchtime purchase is greater than the lunch recommendation of 750 calories. Use \(\alpha=.01\). d. Would it be reasonable to generalize the conclusion of the test in Part (c) to the lunchtime fast-food purchases of all adult Americans? Explain why or why not.e. Explain why it is better to use the customer receipt to determine what was ordered rather than just asking a customer leaving the restaurant what he or she purchased. f. Do you think that asking a customer to provide his or her receipt before they ordered could have introduced a potential bias? Explain.

A researcher speculates that because of differences in diet, Japanese children may have a lower mean blood cholesterol level than U.S. children do. Suppose that the mean level for U.S. children is known to be 170 . Let \(\mu\) represent the mean blood cholesterol level for all Japa-nese children. What hypotheses should the researcher test?

A certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of a writing machine) is at least 10 hours. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting data supports the use of a one-sample \(t\) test. The relevant hypotheses are \(H_{0}: \mu=10\) versus \(H_{a}: \mu<10 .\) a. If \(t=-2.3\) and \(\alpha=.05\) is selected, what conclusion is appropriate?

Occasionally, warning flares of the type contained in most automobile emergency kits fail to ignite. A consumer advocacy group wants to investigate a claim against a manufacturer of flares brought by a person who claims that the proportion of defective flares is much higher than the value of 1 claimed by the manufacturer. A large number of flares will be tested, and the results will be used to decide between \(H_{0}: p=.1\) and \(H_{a}: p>.1\), where \(p\) represents the proportion of defective flares made by this manufacturer. If \(H_{0}\) is rejected, charges of false advertising will be filed against the manufacturer. a. Explain why the alternative hypothesis was chosen to be \(H_{a}: p>.1\). b. In this context, describe Type I and Type II errors, and discuss the consequences of each.
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