In a study of 420,095 Danish cell phone users, 135 subjects...

In a study of 420,095 Danish cell phone users, 135 subjects developed cancer of the brain of nervous system (based on data from the Journal of the National Cancer Institute as reported in USA Today. Test the claim of a once popular belief that such cancers are affected by cell phone use. That is, test the claim that cell phone users develop cancer of the brain or nervous system at a rate that is different from the rate of $0.0340 \%$ for people who do not use cell phones. Because this issue has such great importance, use a 0.005 significance level. Should call phone users be concerned about cancer of the brain or nervous system?

Key Concepts

p-value

The p-value is the probability of obtaining a sample result as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true. It quantifies the evidence against the null hypothesis. A p-value smaller than the significance level indicates strong evidence against the null hypothesis, supporting the alternative hypothesis.

Test of Proportion

The test of proportion is a statistical test used to compare an observed proportion with a hypothesized proportion. It is particularly useful when examining the rate of occurrence of an event in a sample relative to a known or assumed rate in a larger population. This method is applied when evaluating whether the observed incidence of an outcome, such as a disease, deviates significantly from an expected rate.

Type I Error

Type I error occurs when the null hypothesis is incorrectly rejected even though it is true. In hypothesis testing, this error is controlled by setting a significance level, which limits the probability of making such an error. Maintaining a low significance level reduces the risk of concluding that there is an effect when none actually exists.

Null Hypothesis (H?) and Alternative Hypothesis (H?)

The null hypothesis is the assumption that there is no difference or effect—in this case, that the proportion of cancer cases among cell phone users is equal to the known rate for non-users. The alternative hypothesis posits that the cancer rate among cell phone users is different from the established rate. This framework is fundamental in comparing observed data against an expected baseline.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating a null hypothesis, which represents the status quo or no effect, and an alternative hypothesis, which represents a different state or effect. The testing process then seeks to determine whether there is enough evidence in the sample data to reject the null hypothesis in favor of the alternative.

Significance Level

The significance level is the probability threshold at which the null hypothesis is rejected. It represents the risk of making a Type I error, which is rejecting a true null hypothesis. A significance level of 0.005 indicates a very strict criterion, meaning that there is only a 0.5% chance of incorrectly rejecting the null hypothesis.

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