Using Technology
In Exercises 5–8, identify the indicated values or interpret the given display. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section. Use a 0.05 significance level and answer the following:


a. Is the test two-tailed, left-tailed, or right-tailed?
b. What is the test statistic?
c. What is the P-value?
d. What is the null hypothesis, and what do you conclude about it?
e. What is the final conclusion?


Adverse Reactions to Drug The drug Lipitor (atorvastatin) is used to treat high cholesterol. In a clinical trial of Lipitor, 47 of 863 treated subjects experienced headaches (based on data from Pfizer). The accompanying TI-83/84 Plus calculator display shows results from a test of the claim that fewer than 10% of treated subjects experience headaches.

Verified step by step guidance

1. Step 1: Determine the type of test. The problem states that the claim is 'fewer than 10% of treated subjects experience headaches,' which indicates a left-tailed test because we are testing if the proportion is less than a specific value.
2. Step 2: Identify the test statistic. From the TI-83/84 Plus calculator display, the test statistic is z = -4.45929186. This value measures how many standard deviations the sample proportion is from the hypothesized proportion under the null hypothesis.
3. Step 3: Find the P-value. The calculator display shows p = 4.1151493e-6, which is the P-value. This represents the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
4. Step 4: State the null hypothesis. The null hypothesis (H₀) is that the proportion of treated subjects experiencing headaches is equal to or greater than 10% (p ≥ 0.1). The alternative hypothesis (H₁) is that the proportion is less than 10% (p < 0.1).
5. Step 5: Draw a conclusion. Since the P-value is much smaller than the significance level of 0.05, we reject the null hypothesis. This means there is sufficient evidence to support the claim that fewer than 10% of treated subjects experience headaches.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0. In this case, the null hypothesis states that the proportion of subjects experiencing headaches is 10% or more, while the alternative hypothesis suggests it is less than 10%.

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of observing the test results, or something more extreme, assuming that the null hypothesis is true. A smaller P-value indicates stronger evidence against H0. In this scenario, the P-value is extremely low (approximately 4.115e-6), suggesting strong evidence to reject the null hypothesis.

Test Statistic

The test statistic is a standardized value that is calculated from sample data during a hypothesis test. It measures how far the sample statistic is from the null hypothesis value, expressed in terms of standard deviations. In this case, the test statistic (z) is -4.459, indicating that the sample proportion of subjects experiencing headaches is significantly lower than the hypothesized proportion of 10%.