Emergency Room The proportion of patients who visit the emergency room (ER) and die within the year is 0.05. Source: SuperFreakonomics. Suppose a hospital administrator is concerned that his ER has a higher proportion of patients who die within the year. In a random sample of 250 patients who have visited the ER in the past year, 17 have died. Should the administrator be concerned?

Verified step by step guidance

1. Identify the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \). Here, \( H_0: p = 0.05 \) (the proportion of patients who die is 0.05) and \( H_a: p > 0.05 \) (the proportion is higher than 0.05).
2. Calculate the sample proportion \( \hat{p} \) using the data: \( \hat{p} = \frac{17}{250} \).
3. Compute the standard error (SE) of the sampling distribution under the null hypothesis using the formula: \( SE = \sqrt{\frac{p(1-p)}{n}} \) where \( p = 0.05 \) and \( n = 250 \).
4. Calculate the test statistic (z-score) using the formula: \( z = \frac{\hat{p} - p}{SE} \) This measures how many standard errors the sample proportion is away from the hypothesized proportion.
5. Determine the p-value corresponding to the calculated z-score for a right-tailed test. Compare the p-value to a significance level (commonly \( \alpha = 0.05 \)) to decide whether to reject \( H_0 \) and conclude if the administrator should be concerned.

Key Concepts

Hypothesis Testing for Proportions

Hypothesis testing for proportions involves assessing whether the observed sample proportion significantly differs from a known or claimed population proportion. It uses a null hypothesis (e.g., the ER death rate is 0.05) and an alternative hypothesis (e.g., the ER death rate is higher) to determine if the sample data provides enough evidence to reject the null.

Sampling Distribution of a Proportion

The sampling distribution of a sample proportion describes how the proportion varies from sample to sample. For large samples, it approximates a normal distribution with mean equal to the population proportion and standard error calculated from the population proportion and sample size, enabling probability calculations.

Significance Level and p-value

The significance level (commonly 0.05) is the threshold for deciding whether to reject the null hypothesis. The p-value measures the probability of observing a sample proportion as extreme as the one obtained, assuming the null is true. A p-value less than the significance level indicates strong evidence against the null.