Questions: A recent study examined hearing loss data for 1719 U.S. teenagers. In this sample, 349 were found to have some level of hearing loss. News of this study spread quickly, with many news articles blaming the prevalence of hearing loss on the higher use of ear buds by teens. At MSNBC.com (8/17/2010), Carla Johnson summarized the study with the headline: " 1 in 5 U.S. teens has hearing loss, study says." To investigate whether this is an appropriate or a misleading headline, you will conduct a test of significance with the following hypotheses: Null: π=0.20 Alternative: π ≠ 0.20 Use the Theory-Based Inference applet to determine a p-value. Round your answer to 4 decimal places, e.g. 0.7534.

Solution Steps

We are given the following hypotheses:

• Null Hypothesis (\(H_0\)): \(\pi = 0.20\)
• Alternative Hypothesis (\(H_1\)): \(\pi \neq 0.20\)

The sample size (\(n\)) is 1719, and the number of teenagers with hearing loss is 349. The sample proportion (\(\hat{p}\)) is calculated as: \[ \hat{p} = \frac{349}{1719} = 0.2030 \]

The test statistic for the hypothesis test is given by: \[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Substituting the values: \[ Z = \frac{0.2030 - 0.20}{\sqrt{\frac{0.20 \times 0.80}{1719}}} = 0.3135 \]

The P-value associated with the test statistic \(Z = 0.3135\) is 0.7539.

For a two-tailed test at the significance level \(\alpha = 0.05\), the critical region is: \[ Z < -1.96 \quad \text{or} \quad Z > 1.96 \]

Since the test statistic \(Z = 0.3135\) does not fall within the critical region and the P-value \(0.7539\) is greater than \(\alpha = 0.05\), we fail to reject the null hypothesis.

Final Answer