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Chapter 10: Problem 23

Use the definition of the \(P\) -value to explain the following: a. Why \(H_{0}\) would be rejected if \(P\) -value \(=.0003\) b. Why \(H_{0}\) would not be rejected if \(P\) -value \(=.350\)

Short Answer

Expert verified
The P-value quantifies the strength of evidence against the null hypothesis \(H_{0}\). If the P-value is below a predetermined significance level (often 0.05), then \(H_{0}\) is rejected. Thus, for a P-value of 0.0003 (< 0.05), \(H_{0}\) would be rejected. Conversely, for a P-value of 0.350 (> 0.05), there is not sufficient evidence to reject \(H_{0}\), so \(H_{0}\) is not rejected.

Step by step solution

01

Understanding the P-value

The P-value in a hypothesis testing scenario represents the probability of obtaining a test statistic result at least as extreme as the one that was actually observed, given that the null hypothesis is true. If this probability (the P-value) is smaller than a predetermined significance level (typically 0.05), then there is strong evidence against the null hypothesis, so it is rejected.
02

Application of P-value interpretation to part a

When the P-value is 0.0003, it means there's only a 0.03% chance of obtaining a test statistic as extreme as the observed one, assuming the null hypothesis is true. This is a very low probability, far below the typical 5% significance level. As such, the evidence against the null hypothesis is considered strong, and hence \(H_{0}\) would be rejected.
03

Application of P-value interpretation to part b

On the other hand, when the P-value is 0.350, it means there's a 35% chance of obtaining a test statistic as extreme as the observed one if the null hypothesis is true. This is a far higher probability, above the common 5% significance level. Therefore, the evidence against the null hypothesis is considered not strong enough to reject it, so \(H_{0}\) would not be rejected.

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Key Concepts

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

Null Hypothesis Rejection
The null hypothesis, often denoted by H_{0}, is a general statement or default position that there is no relationship between two measured phenomena or no association among groups. Rejecting the null hypothesis is a crucial step in hypothesis testing. It essentially means that you have sufficient evidence to conclude that the null hypothesis is not likely to be true.

In the context of the exercise, if the P-value is .0003, it indicates that the results observed in the study (or more extreme results) would occur only 0.03% of the time if the null hypothesis were true. Since this chance is extremely low, scientists have agreed on a threshold below which they conclude the null hypothesis is unlikely. This leads to its rejection, and it's typically done with a high level of confidence in the result.
Significance Level
The significance level, denoted by \(\beta\), is a threshold chosen by the researcher to decide whether to reject the null hypothesis. The most commonly used significance level is 0.05 or 5%. This standard has been established as a balance between being too lenient and too strict when it comes to identifying a statistically significant result.

The significance level is a critical value in statistical tests and represents the probability of rejecting the null hypothesis when it is actually true, an error known as a Type I error. When your P-value is lower than the significance level, it implies that the observed data are inconsistent with the assumption that the null hypothesis is true, favoring the alternative hypothesis.
Test Statistic
A test statistic is a value calculated from the sample data during a hypothesis test. Its purpose is to help determine the likelihood of the null hypothesis given the data. The test statistic can be a t-value, z-score, chi-squared statistic, or any number of other statistics, depending on the type of test being performed.

In practical terms, the test statistic compares your data to what is expected under the null hypothesis. The more the test statistic diverges from what we'd expect if \(H_{0}\) were true, the less likely the null hypothesis is accurate. The P-value directly relates to the test statistic, as it tells you the probability of seeing a value as extreme as or more extreme than the test statistic, assuming the null hypothesis is true.
Probability
In hypothesis testing, probability measures the likelihood of a certain event occurring. It ranges from 0 to 1, with lower values indicating a lower likelihood and higher values indicating a greater likelihood. When you calculate a P-value in the context of hypothesis testing, you're computing the probability of obtaining results at least as extreme as the observed results, under the assumption that the null hypothesis is correct.

The concept of probability is foundational when interpreting P-values. A P-value of .350 means there is a 35% chance of obtaining a test statistic as extreme as the observed one if the null hypothesis were true鈥攁 relatively high probability, which suggests that such an outcome isn't particularly unusual and, thus, doesn't provide strong evidence against the null hypothesis.

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Most popular questions from this chapter

Explain why the statement \(\bar{x}=50\) is not a legitimate hypothesis.

The paper "MRI Evaluation of the Contralateral Breast in Women with Recently Diagnosed Breast Cancer" (New England Journal of Medicine \([2007]: 1295-1303)\) describes a study of the use of MRI (Magnetic Resonance Imaging) exams in the diagnosis of breast cancer. The purpose of the study was to determine if MRI exams do a better job than mammograms of determining if women who have recently been diagnosed with cancer in one breast have cancer in the other breast. The study participants were 969 women who had been diagnosed with cancer in one breast and for whom a mammogram did not detect cancer in the other breast. These women had an MRI exam of the other breast, and 121 of those exams indicated possible cancer. After undergoing biopsies, it was determined that 30 of the 121 did in fact have cancer in the other breast, whereas 91 did not. The women were all followed for one year, and three of the women for whom the MRI exam did not indicate cancer in the other breast were subsequently diagnosed with cancer that the MRI did not detect. The accompanying table summarizes this information. Suppose that for women recently diagnosed with cancer in only one breast, the MRI is used to decide between the two "hypotheses" \(H_{0}\) : woman has cancer in the other breast \(H_{a}:\) woman does not have cancer in the other breast (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) a. One possible error would be deciding that a woman who does have cancer in the other breast is cancerfree. Is this a Type I or a Type II error? Use the information in the table to approximate the probability of this type of error. b. There is a second type of error that is possible in this setting. Describe this error and use the information in the given table to approximate the probability of this type of error.

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 bpm and 15 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 \(\alpha=.01\). c. Based on the outcomes of the tests in Parts (a) and (b), write a paragraph comparing the benefits of treadmill walking and Wii Bowling in terms of raising heart rate over the resting heart rate.

A certain university has decided to introduce the use of plus and minus with letter grades, as long as there is evidence that more than \(60 \%\) of the faculty favor the change. A random sample of faculty will be selected, and the resulting data will be used to test the relevant hypotheses. If \(p\) represents the proportion of all faculty that favor a change to plus-minus grading, which of the following pair of hypotheses should the administration test: $$ H_{0}: p=.6 \text { versus } H_{a}: p<.6 $$ or $$ H_{0}: p=.6 \text { versus } H_{a}: p>.6 $$ Explain your choice.

Let \(\mu\) denote the true average lifetime (in hours) for a certain type of battery under controlled laboratory conditions. A test of \(H_{0}: \mu=10\) versus \(H_{a}\) : \(\mu<10\) will be based on a sample of size \(36 .\) Suppose that \(\sigma\) is known to be 0.6 , from which \(\sigma_{\bar{x}}=.1\). The appropriate test statistic is then $$ z=\frac{\bar{x}-10}{0.1} $$ a. What is \(\alpha\) for the test procedure that rejects \(H_{0}\) if \(z \leq-1.28 ?\) b. If the test procedure of Part (a) is used, calculate \(\beta\) when \(\mu=9.8\), and interpret this error probability. c. Without doing any calculation, explain how \(\beta\) when \(\mu=9.5\) compares to \(\beta\) when \(\mu=9.8\). Then check your assertion by computing \(\beta\) when \(\mu=9.5\). d. What is the power of the test when \(\mu=9.8\) ? when \(\mu=9.5 ?\)
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