Sleeping Patterns of Pregnant Women A random sample of 150 pregnant women indicated that 81 napped at least twice per week. Do a majority of pregnant women nap at least twice a week? Use the α = 0.05 level of significance.


Source: National Sleep Foundation.

Verified step by step guidance

1. Identify the parameter of interest: the population proportion \( p \) of pregnant women who nap at least twice per week.
2. Set up the null and alternative hypotheses. The null hypothesis \( H_0 \) assumes no majority, so \( p = 0.5 \). The alternative hypothesis \( H_a \) tests if the majority nap at least twice per week, so \( p > 0.5 \). Formally: \( H_0: p = 0.5 \) and \( H_a: p > 0.5 \).
3. Calculate the sample proportion \( \hat{p} \) using the data: \( \hat{p} = \frac{81}{150} \).
4. Compute the test statistic for a one-proportion z-test using the formula: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \] where \( p_0 = 0.5 \) and \( n = 150 \).
5. Determine the critical value from the standard normal distribution for \( \alpha = 0.05 \) in a right-tailed test, then compare the calculated \( z \)-value to this critical value to decide whether to reject \( H_0 \).

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves formulating a null hypothesis (no effect or status quo) and an alternative hypothesis, then using sample evidence to decide whether to reject the null hypothesis at a given significance level.

Significance Level (α)

The significance level, denoted by α, is the threshold probability for rejecting the null hypothesis. It represents the risk of making a Type I error, which is rejecting a true null hypothesis. Common values are 0.05 or 5%, meaning there is a 5% chance of incorrectly concluding an effect exists.

Proportion Testing

Proportion testing involves assessing whether the observed sample proportion differs significantly from a hypothesized population proportion. This is often done using a z-test for proportions, comparing the sample proportion to a claimed value to determine if the difference is statistically significant.