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Chapter 8: Problem 25

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section. Through the sample of the first 49 Super Bowls, 28 of them were won by teams in the National Football Conference (NFC). Use a 0.05 significance level to test the claim that the probability of an NFC team Super Bowl win is greater than one-half.

Short Answer

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01

= -1.9522

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Key Concepts

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

null hypothesis
In hypothesis testing, the null hypothesis is a statement that indicates no effect or no difference. It is a starting assumption for statistical tests. For example, if we are testing the effect of a new drug, the null hypothesis would state that the drug has no effect compared to a placebo. This statement is typically denoted as \( H_0 \). For the exercise above, the null hypothesis is that the proportion of Super Bowl wins by NFC teams is equal to one-half (i.e., there's no difference from a 50% win rate). This is formally written as:
\[ H_0: p = 0.5 \]
This hypothesis is tested to see if there is sufficient evidence to reject it in favor of the alternative hypothesis.
alternative hypothesis
The alternative hypothesis is the statement that we are trying to find evidence for. It is denoted as \( H_1 \) or \( H_a \). This hypothesis represents a new effect or difference that is of interest to the researcher. In the given problem, we want to test if the probability of an NFC team winning the Super Bowl is greater than one-half, indicating that NFC teams win more frequently than AFC teams. This is formally written as:
\[ H_1: p > 0.5 \]
The alternative hypothesis is accepted if the evidence against the null hypothesis is strong, based on the sample data.
P-value
The P-value helps us determine the strength of the evidence against the null hypothesis. It is the probability of observing test results at least as extreme as the ones observed, under the assumption that the null hypothesis is true. A small P-value (typically 鈮� 0.05) indicates strong evidence against the null hypothesis, leading us to reject it. Conversely, a large P-value suggests weak evidence against the null hypothesis, so we fail to reject it. In our Super Bowl example, the P-value would tell us how probable it is to observe NFC winning 28 out of 49 times (or more) if the actual win probability is 0.5.
normal distribution
The normal distribution is a continuous probability distribution that is symmetrical and bell-shaped. Many statistical tests, including the one in our exercise, assume that the sample data approximates a normal distribution, especially when the sample size is large. In hypothesis tests, we often use the normal distribution as a model to derive critical values and P-values. For the Super Bowl example, since the sample size is comparatively small (49 games), the binomial distribution can be approximated by the normal distribution due to the central limit theorem.
significance level
The significance level, denoted as \( \alpha \), is a threshold that determines when to reject the null hypothesis. It represents the probability of making a Type I error, which is the incorrect rejection of a true null hypothesis. Common significance levels are 0.05, 0.01, and 0.10. For our Super Bowl problem, the significance level is set to 0.05. This means that we would reject the null hypothesis if our results have a less than 5% probability of occurring under the null hypothesis. Choosing the significance level is crucial, as it balances the risk of Type I errors against the power of the test.

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Most popular questions from this chapter

The Ericsson method is one of several methods claimed to increase the likelihood of a baby girl. In a clinical trial, results could be analyzed with a formal hypothesis test with the alternative hypothesis of \(p>0.5,\) which corresponds to the claim that the method increases the likelihood of having a girl, so that the proportion of girls is greater than 0.5. If you have an interest in establishing the success of the method, which of the following P-values would you prefer: \(0.999,0.5,0.95,0.05,0.01,0.001 ?\) Why?

Assume that a simple random sample has been selected and test the given claim. Unless specified by your instructor, use either the P-value method or the critical value method for testing hypotheses. Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), or critical value(s), and state the final conclusion that addresses the original claim. In a test of the effectiveness of garlic for lowering cholesterol, 49 subjects were treated with raw garlic. Cholesterol levels were measured before and after the treatment. The changes (before minus after) in their levels of LDL cholesterol (in \(\mathrm{mg} / \mathrm{dL}\) ) have a mean of 0.4 and a standard deviation of 21.0 (based on data from "Effect of Raw Garlic vs Commercial Garlic Supplements on Plasma Lipid Concentrations in Adults with Moderate Hypercholesterolemia," by Gardner et al., Archives of Internal Medicine, Vol. \(167,\) No. 4 ). Use a 0.05 significance level to test the claim that with garlic treatment, the mean change in LDL cholesterol is greater than \(0 .\) What do the results suggest about the effectiveness of the garlic treatment?

Use a significance level of \(\alpha=0.05\) and use the given information for the following: a. State a conclusion about the null hypothesis. (Reject \(H_{0}\) or fail to reject \(H_{0 .}\) ) b. Without using technical terms or symbols, state a final conclusion that addresses the original claim. Original claim: Fewer than \(90 \%\) of adults have a cell phone. The hypothesis test results in a \(P\) -value of 0.0003.

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population. Fast Food Drive-Through Service Times Listed below are drive-through service times (seconds) recorded at McDonald's during dinner times (from Data Set 25 "Fast Food" in Appendix B). Assuming that dinner service times at Wendy's have standard deviation \(\sigma=55.93\) sec, use a 0.01 significance level to test the claim that service times at McDonald's have the same variation as service times at Wendy's. Should McDonald's take any action? $$\begin{array}{cccccccc} 121 & 119 & 146 & 266 & 333 & 308 & 333 & 308 \end{array}$$

Identifying \(H_{0}\) and \(H_{1}\) Do the following: a. Express the original claim in symbolic form. b. Identify the null and alternative hypotheses. Claim: Fewer than \(95 \%\) of adults have a cell phone. In a Marist poll of 1128 adults, \(87 \%\) said that they have a cell phone.
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