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Chapter 10: Problem 24

For which of the following \(P\) -values will the null hypothesis be rejected when performing a test with a significance level of .05: a. .001 d. .047 b. .021 e. .148 c. .078

Short Answer

Expert verified
For \(P\) values of .001, .047, and .021, the null hypothesis would be rejected when performing a test with a significance level of .05.

Step by step solution

01

Compare the first \(P\) value with the significance level

The first provided \(P\) value is .001. As .001 is less than the significance level of .05, for this value of \(P\), the null hypothesis would be rejected.
02

Compare the second \(P\) value with the significance level

The second provided \(P\) value is .047. As .047 is less than the significance level of .05, for this value of \(P\), the null hypothesis would also be rejected.
03

Compare the third \(P\) value with the significance level

The third provided \(P\) value is .021. As .021 is also less than the significance level of .05, for this value of \(P\), again, the null hypothesis would be rejected.
04

Compare the fourth \(P\) value with the significance level

The fourth provided \(P\) value is .148. But this time, .148 is greater than the significance level of .05. So, for this value of \(P\), the null hypothesis would not be rejected.
05

Compare the fifth \(P\) value with the significance level

The fifth and last provided \(P\) value is .078. As .078 is greater than the significance level of .05, for this value of \(P\), the null hypothesis would not be rejected either. Thus, resulting in a total of three \(P\) values (.001, .047, .021) where the null hypothesis would be rejected for a significance level of .05.

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Key Concepts

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

P-value
The P-value is a probability that helps determine the strength of the results from a hypothesis test. Essentially, it tells us how likely it is to observe the test results under the assumption that the null hypothesis is true. When analyzing P-values, the smaller the number, the stronger the evidence against the null hypothesis.
  • If the P-value is small (typically less than the chosen significance level), it suggests that the observed data is unlikely under the null hypothesis.
  • Conversely, a large P-value indicates that the observed data is not surprising under the null hypothesis.
It's like flipping a coin鈥攊f you expect it to be fair (50% heads and 50% tails) but see it land on heads every time, a low P-value would suggest the coin could be biased.
null hypothesis
In hypothesis testing, the null hypothesis ( \(H_0\)) is a statement that indicates no effect or no difference. It is the default position that there is nothing unusual happening.
  • The null hypothesis acts as a starting point for statistical testing.
  • It's said to be 'accepted' if the test results show insufficient evidence against it, meaning the P-value is higher than the significance level (alpha).
For instance, if we are testing a new drug, the null hypothesis would state that the drug has no effect compared to the existing treatment. The main goal is to use sample data to determine if this hypothesis can be rejected.
significance level (alpha)
The significance level, often denoted by alpha ( \(\alpha\)), is a threshold set by the researcher which defines how much evidence is required to reject the null hypothesis.
  • A common significance level is 0.05, but it can be adjusted based on the researcher's criteria and the context of the test.
  • If the P-value is less than or equal to the significance level, it suggests that the evidence is strong enough to reject the null hypothesis.
Choosing an appropriate alpha level involves balancing the risk of making a Type I error (rejecting the null hypothesis when it is true) against the need to detect a true effect. Setting a lower alpha level means more evidence is needed to reject the null hypothesis, reducing the risk of falsely detecting an effect.

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Most popular questions from this chapter

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 \(150^{\circ} \mathrm{F}\), a scientist will take 50 water samples at randomly selected times and will record the water temperature of each sample. She will then use a \(z\) statistic $$ z=\frac{\bar{x}-150}{\frac{\sigma}{\sqrt{n}}} $$ to decide between the hypotheses \(H_{0}: \mu=150\) and \(H_{a}: \mu>150,\) where \(\mu\) is the mean temperature of discharged water. Assume that \(\sigma\) is known to be 10 . a. Explain why use of the \(z\) statistic is appropriate in this setting. b. Describe Type I and Type II errors in this context. \(c\). The rejection of \(H_{0}\) when \(z \geq 1.8\) corresponds to what value of \(\alpha\) ? (That is, what is the area under the \(z\) curve to the right of \(1.8 ?\) ) d. Suppose that the actual value for \(\mu\) is 153 and that \(H_{0}\) is to be rejected if \(z \geq 1.8 .\) Draw a sketch (similar to that of Figure 10.5 ) of the sampling distribution of \(\bar{x},\) and shade the region that would represent \(\beta\), the probability of making a Type II error. e. For the hypotheses and test procedure described, compute the value of \(\beta\) when \(\mu=153\). f. For the hypotheses and test procedure described, what is the value of \(\beta\) if \(\mu=160\) ? g. What would be the conclusion of the test if \(H_{0}\) is rejected when \(z \geq 1.8\) and \(\bar{x}=152.4\) ? What type of error might have been made in reaching this conclusion?

In a survey of 1005 adult Americans, \(46 \%\) indicated that they were somewhat interested or very interested in having web access in their cars (USA Today, May I. 2009 ). Suppose that the marketing manager of a car manufacturer claims that the \(46 \%\) is based only on a sample and that \(46 \%\) is close to half, so there is no reason to believe that the proportion of all adult Americans who want car web access is less than \(.50 .\) Is the marketing manager correct in his claim? Provide statistical evidence to support your answer. For purposes of this exercise, assume that the sample can be considered as representative of adult Americans.

A manufacturer of hand-held calculators receives large shipments of printed circuits from a supplier. It is too costly and time-consuming to inspect all incoming circuits, so when each shipment arrives, a sample is selected for inspection. Information from the sample is then used to test \(H_{0}: p=.01\) versus \(H_{a}: p>.01\), where \(p\) is the actual proportion of defective circuits in the shipment. If the null hypothesis is not rejected, the shipment is accepted, and the circuits are used in the production of calculators. If the null hypothesis is rejected, the entire shipment is returned to the supplier because of inferior quality. (A shipment is defined to be of inferior quality if it contains more than \(1 \%\) defective circuits.) a. In this context, define Type I and Type II errors. b. From the calculator manufacturer's point of view, which type of error is considered more serious? c. From the printed circuit supplier's point of view, which type of error is considered more serious?

Do state laws that allow private citizens to carry concealed weapons result in a reduced crime rate? The author of a study carried out by the Brookings Institution is reported as saying, "The strongest thing I could say is that I don't see any strong evidence that they are reducing crime" (San Luis Obispo Tribune, January 23 . 2003 ). a. Is this conclusion consistent with testing \(H_{0}:\) concealed weapons laws reduce crime versus \(H_{a}:\) concealed weapons laws do not reduce crime or with testing \(H_{0}:\) concealed weapons laws do not reduce crime versus \(H_{a}\) : concealed weapons laws reduce crime Explain. b. Does the stated conclusion indicate that the null hypothesis was rejected or not rejected? Explain.

Paint used to paint lines on roads must reflect enough light to be clearly visible at night. Let \(\mu\) denote the mean reflectometer reading for a new type of paint under consideration. A test of \(H_{0}: \mu=20\) versus \(H_{a}: \mu>20\) based on a sample of 15 observations gave \(t=3.2\). What conclusion is appropriate at each of the following significance levels? a. \(\quad \alpha=.05\) c. \(\alpha=.001\) b. \(\quad \alpha=.01\)
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