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Chapter 4: Problem 40

A study of 420,095 Danish cell phone users resulted in 135 who developed cancer of the brain or nervous system (based on data from the Journal of the National Cancer Institute). When comparing this sample group to another group of people who did not use cell phones, it was found that there is a probability of \(0.512\) of getting such sample results by chance. What do you conclude?

Short Answer

Expert verified
Since the p-value is higher than 0.05, we fail to reject the null hypothesis. There is not enough evidence to support that cell phone usage increases the risk of cancer.

Step by step solution

01

Understand the Hypotheses

Identify the null hypothesis (H_0) and the alternative hypothesis (H_a). The null hypothesis states that there is no effect or no difference. In this case, it means cell phone usage does not affect the probability of developing cancer. The alternative hypothesis states that there is an effect or a difference, meaning cell phone usage does affect the probability of developing cancer.
02

Determine the Significance Level

Typically, the significance level (伪) is set at 0.05, representing a 5% risk of concluding that a difference exists when there is no actual difference. However, verify the 伪 level given in the exercise or implied by context. If not given, assume 伪 = 0.05.
03

Compare p-value with Significance Level

The problem states the p-value is 0.512. Compare this p-value to the significance level (伪). If p-value > 伪, fail to reject the null hypothesis. If p-value 鈮� 伪, reject the null hypothesis.
04

Draw Conclusion

Since the p-value (0.512) is greater than the significance level (0.05), we fail to reject the null hypothesis. There is insufficient evidence to conclude that cell phone usage is associated with an increased risk of developing cancer of the brain or nervous system.

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Key Concepts

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

Null Hypothesis
In statistical hypothesis testing, the null hypothesis (denoted as H0) represents a statement of no effect or no difference. It assumes that any observed effect or difference in a study arises purely due to chance. Here, in the context of studying cell phone users and cancer development, the null hypothesis is that cell phone usage does not affect the probability of developing brain or nervous system cancer. Formulating the null hypothesis is the first step in any hypothesis testing process.
Alternative Hypothesis
Opposite to the null hypothesis is the alternative hypothesis (denoted as Ha or H1). This hypothesis suggests that there is a statistically significant effect or difference. In our study, the alternative hypothesis would be that cell phone usage does affect the probability of developing brain or nervous system cancer. The goal of hypothesis testing is to determine whether the data provide sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.
Significance Level
The significance level (denoted as 伪) is a threshold set by the researcher which determines the probability of rejecting the null hypothesis when it is actually true (a type I error). Commonly, a significance level of 0.05 is used, meaning there is a 5% risk of concluding that there is a difference when there isn't one in reality. In the given exercise, if the significance level is 0.05, it allows us to make decisions about whether observed results are statistically significant.
P-value
The p-value is a measure that helps us determine the strength of the evidence against the null hypothesis. It indicates the probability of observing the given results, or more extreme ones, if the null hypothesis is true. In our example, the p-value is 0.512. This relatively high p-value suggests that the observed result of 135 cancer cases among cell phone users is quite likely under the null hypothesis. By comparing the p-value with the significance level, we decide whether or not to reject the null hypothesis.
Statistical Conclusion
After comparing the p-value to the significance level, we make a statistical conclusion. If the p-value is 鈮� significance level, we reject the null hypothesis; if the p-value is > significance level, we fail to reject the null hypothesis. In the given study, the p-value of 0.512 is much higher than the significance level of 0.05. Therefore, we fail to reject the null hypothesis, concluding that there is insufficient evidence to suggest that cell phone usage increases the risk of brain or nervous system cancer. This conclusion helps us to understand that, based on the data, there doesn't appear to be a link between cell phone usage and cancer risk.

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Most popular questions from this chapter

Express all probabilities as fractions. In a horse race, a quinela bet is won if you selected the two horses that finish first and second, and they can be selected in any order. The 140 th running of the Kentucky Derby had a field of 19 horses. What is the probability of winning a quinela bet if random horse selections are made?

Find the probability. It has been reported that \(20 \%\) of iPhones manufactured by Foxconn for a product launch did not meet Apple's quality standards. An engineer needs at least one defective iPhone so she can try to identify the problem(s). If she randomly selects 15 iPhones from a very large batch, what is the probability that she will get at least 1 that is defective? Is that probability high enough so that she can be reasonably sure of getting a defect for her work?

A problem that once attracted much attention is the Monty Hall problem, based on the old television game show Let's Make a Deal, hosted by Monty Hall. Suppose you are a contestant who has selected one of three doors after being told that two of them conceal nothing but that a new red Corvette is behind one of the three. Next, the host opens one of the doors you didn't select and shows that there is nothing behind it. He then offers you the choice of sticking with your first selection or switching to the other unopened door. Should you stick with your first choice or should you switch? Develop a simulation of this game and determine whether you should stick or switch. (According to Chance magazine, business schools at such institutions as Harvard and Stanford use this problem to help students deal with decision making.)

A classical probability problem involves a king who wanted to increase the proportion of women by decreeing that after a mother gives birth to a son, she is prohibited from having any more children. The king reasons that some families will have just one boy, whereas other families will have a few girls and one boy, so the proportion of girls will be increased. Is his reasoning correct? Will the proportion of girls increase?

Express the indicated degree of likelihood as a probability value between 0 and \(1 .\) If you make a random guess for the answer to a true/false test question, there is a \(50-50\) chance of being correct.
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