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Chapter 10: Problem 69

The amount of shaft wear after a fixed mileage was determined for each of 7 randomly selected internal combustion engines, resulting in a mean of \(0.0372\) in. and a standard deviation of \(0.0125 \mathrm{in}\). 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 \(\mu=.04\) ? c. What is the approximate power of the test when \(\mu=\) \(.04\) and \(\alpha=.05\) ?

Short Answer

Expert verified
The solution to parts a, b, and c requires calculation and interpretation of the test statistic, the probability of a Type II Error, and the power of the test, respectively. The exact numeric answers will depend on the calculated z-scores and corresponding areas under the standard normal curve.

Step by step solution

01

Hypothesis Testing

Hypotheses being tested are \(H_{0}: \mu=.035\) versus \(H_{a}:\) \(\mu>.035\). Given that the mean \(\bar {X}=0.0372\), the standard deviation \( \sigma =0.0125\), the number of samples \(n=7\) and the significance level \(\alpha=0.05\). First, compute the test statistic (z-score) using the formula \(z=\frac{\bar {X}-\mu_{0}}{\sigma/\sqrt{n}}\), where \(\mu_{0}\) represents the hypothesized population mean under the null hypothesis (i.e. 0.035). Next, the p-value associated with this observed value of the test statistic is determined by looking up the z-score in a standard normal distribution table. If the p-value is less than or equal to 0.05, the null hypothesis is rejected. If the p-value is greater than 0.05, there is insufficient evidence to reject the null hypothesis.
02

Type II Error

The probability of a Type II error (\(\beta\)) is the probability of failing to reject the null hypothesis when it is false. First, calculate the critical value of z (\(z_{\alpha}\)) from the standard normal distribution table using \(1-\alpha\). Then, calculate the z-score for \(\mu=.04\) using the formula \(z=\frac{\mu-\mu_{0}}{\sigma/\sqrt{n}}\). Lastly, the value of \(\beta\) is estimated from the standard normal distribution table.
03

Test Power

The power of the test is the probability of rejecting the null hypothesis when it is false, which is \(1-\beta\). Given that the value of \(\beta\) was obtained in the previous step, simply subtract \(\beta\) from 1 to find the power of the test when \(\mu=.04\) and \(\alpha=.05\).

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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 wrongly reject a true null hypothesis. Think of it like a false alarm. If you conclude that the engine shaft wear is greater than 0.035 inches, when in reality it isn't, that's a Type I error.
Here's an easy way to remember:
  • The null hypothesis is like you're innocent until proven guilty.
  • A Type I error means calling someone guilty when they're actually innocent.
  • In hypothesis testing, the probability of making a Type I error is denoted by \( \alpha \).

In the context of this problem, the significance level \( \alpha \) is set at 0.05, meaning there's a 5% risk of committing this error. Always set \( \alpha \) before conducting a test so you know your tolerance for this risk.
Type II Error
Now, let's understand Type II error, which is kind of the opposite. This occurs when you fail to reject a false null hypothesis, meaning you missed detecting something that's actually there.
In simple terms:
  • Imagine your fire alarm doesn't ring when there's really a fire.
  • The null hypothesis is wrong, but you didn't reject it.
  • Type II errors have a probability represented by \( \beta \).

So, what \( \beta \) tells us is how likely we are to not find a problem (shaft wear being more than 0.035 inches) when it truly exists. Minimizing \( \beta \) is important for a more accurate test. It鈥檚 crucial to calculate \( \beta \), especially when the test power is significant, as done in this example for a true mean of 0.04 inches.
Test Power
The power of a test is all about effectiveness. It measures how good you are at detecting a true effect when it exists. In technical terms, it's the probability of rejecting a false null hypothesis, denoted by \( 1-\beta \).
The higher the test power, the better:
  • Power indicates your test's ability to "catch" actual significant differences.
  • A powerful test will detect even small differences if they are real.
  • In this context, realizing the shaft wear is indeed more than supposed can prevent mechanical issues.

For the given problem, if \( \beta \) is calculated and found, power is easily derived. Increasing the test's power might mean adjusting sample size, level of significance, or effect size adjustments, amongst others.
Normal Distribution
Commonly known as the bell curve, normal distribution is at the heart of many statistical tests, including hypothesis testing. It鈥檚 characterized by a symmetrical shape centered around the mean.
For hypothesis tests:
  • Assuming data follows normal distribution lets you use certain mathematical properties.
  • Most data points hover around the mean, with fewer points as you move away.
  • This spread of data is controlled by standard deviation, a key factor in tests.

In our exercise, we assume shaft wear follows this pattern, allowing us to calculate probabilities and z-scores effectively. This assumption is a foundational part of accurately interpreting test results, especially with small sample sizes like 7 engines.

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Most popular questions from this chapter

A county commissioner must vote on a resolution that would commit substantial resources to the construction of a sewer in an outlying residential area. Her fiscal decisions have been criticized in the past, so she decides to take a survey of constituents to find out whether they favor spending money for a sewer system. She will vote to appropriate funds only if she can be fairly certain that a majority of the people in her district favor the measure. What hypotheses should she test?

The city council in a large city has become concerned about the trend toward exclusion of renters with children in apartments within the city. The housing coordinator has decided to select a random sample of 125 apartments and determine for each whether children are permitted. Let \(\pi\) be the true proportion of apartments that prohibit children. If \(\pi\) exceeds \(.75\), the city council will consider appropriate legislation. a. If 102 of the 125 . sampled apartments exclude renters with children, would a level 05 test lead you to the conclusion that more than \(75 \%\) of all apartments exclude children? b. What is the power of the test when \(\pi=.8\) and \(\alpha=.05\) ?

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes \(n\) is the large-sample \(z\) test appropriate: a. \(H_{0}: \pi=.2, n=25\) b. \(H_{0}: \pi=.6, n=210\) c. \(H_{0}: \pi=.9, n=100\) d. \(H_{0}: \pi=.05, n=75\)

Researchers at the University of Washington and Harvard University analyzed records of breast cancer screening and diagnostic evaluations ("Mammogram Cancer Scares More Frequent than Thought," USA Today, April 16,1998 ). Discussing the benefits and downsides of the screening process, the article states that, although the rate of false-positives is higher than previously thought, if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall but the rate of missed cancers would rise. Suppose that such a screening test is used to decide between a null hypothesis of \(H_{0}:\) no cancer is present and an alternative hypothesis of \(H_{0}:\) cancer is present. (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) a. Would a false-positive (thinking that cancer is present when in fact it is not) be a Type I error or a Type II error? b. Describe a Type I error in the context of this problem, and discuss the consequences of making a Type I error. c. Describe a Type II error in the context of this problem, and discuss the consequences of making a Type II error. d. What aspect of the relationship between the probability of Type I and Type II errors is being described by the statement in the article that if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall but the rate of missed cancers would rise?

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).
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