The drug Lipitor is meant to reduce cholesterol and LDL...

The drug Lipitor is meant to reduce cholesterol and LDL cholesterol. In clinical trials, 19 out of 863 patients taking $10 \mathrm{mg}$ of Lipitor daily complained of flulike symptoms. Suppose that it is known that $1.9 \%$ of patients taking competing drugs complain of flulike symptoms. Is there evidence to conclude that more than $1.9 \%$ of Lipitor users experience flulike symptoms as a side effect at the $\alpha=0.01$ level of significance?

Key Concepts

Hypothesis Testing

Hypothesis testing is a method in statistics used to decide whether there is enough evidence to reject a presumed statement (the null hypothesis) about a population parameter. In this context, it is applied to determine if the rate of side effects for Lipitor users is statistically greater than the established rate for competing drugs.

Null and Alternative Hypotheses

The null hypothesis typically posits that there is no difference or that the parameter is equal to a standard value (in this case, the side effect rate is 1.9% for Lipitor as well), while the alternative hypothesis claims that the parameter is greater than that standard. This sets up a one-tailed test to check for an increase in the proportion of side effects.

Significance Level

The significance level, denoted by alpha (?), is the threshold for deciding whether a result is statistically significant. Here, ?=0.01 means there is only a 1% risk of concluding that Lipitor users exhibit a higher rate of flulike symptoms when in fact they do not, which is a stringent criterion to reduce the probability of a Type I error.

Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion describes how the estimated proportion from a sample is distributed around the true population proportion. Under the null hypothesis and with a sufficiently large sample size, this distribution can be approximated by a normal distribution, which is essential for conducting a z-test for proportions.

Test Statistic (Z-Score for Proportion)

The test statistic in a proportion test is computed by taking the difference between the observed sample proportion and the null hypothesized proportion, then dividing that by the standard error of the proportion. This z-score indicates how many standard deviations the observed proportion is from the hypothesized population proportion.

p-value

The p-value in a hypothesis test is the probability, under the null hypothesis, of obtaining a test statistic at least as extreme as the one observed. In a one-tailed test such as this, a p-value less than the significance level (1%) would provide evidence to reject the null hypothesis in favor of the alternative, indicating that the proportion of Lipitor users with side effects is significantly higher.

Transcript

Please give Ace some feedback