Question

it was found in 2010 that 19% of high school seniors were smoking cigarettes. due to the popularity of vaping, a researcher fears that the percentage of high school seniors who are now smoking/vaping nicotine is higher. in a study, they found 85 out of 400 seniors were vaping/smoking nicotine. is there enough evidence at $\alpha = 0.05$ level to support their concern? recommend you do your complete hypothesis test before looking at choices as otherwise it may be confusing. choose the correct hypothesis test. hypothesis $h_0:p = 0.19$ $h_1:p\
eq0.19$ $\alpha=0.05$ two - tail t - test $d.f.=399$ (use 100) test value $z = \frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}=\frac{0.2125 - 0.19}{\sqrt{\frac{(-2127)(7873)}{400}}}\approx2.10$ decision: we do conclusion: there is enough evidence to support the claim that the percentage of high school seniors who are now smoking/vaping nicotine is higher than 19% because the test statistic is in the critical region.

Explanation:

Step1: State the hypotheses

The null hypothesis \(H_0:p = 0.19\) (the proportion of high - school seniors smoking/ vaping nicotine is \(19\%\)). The alternative hypothesis \(H_1:p>0.19\) (the proportion is higher than \(19\%\)) since the researcher fears the percentage is higher.

Step2: Calculate the sample proportion \(\hat{p}\)

The sample proportion \(\hat{p}=\frac{x}{n}\), where \(x = 85\) (number of successes in the sample) and \(n = 400\) (sample size). So \(\hat{p}=\frac{85}{400}=0.2125\)

Step3: Calculate the test statistic \(z\)

The formula for the test statistic in a one - sample proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Substitute \(p = 0.19\), \(\hat{p}=0.2125\), and \(n = 400\) into the formula:

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Step4: Find the critical value

For a one - tailed test with \(\alpha = 0.05\), the critical value \(z_{\alpha}\) from the standard normal distribution table is \(z_{0.05}=1.645\)

Step5: Make a decision

Since the test statistic \(z\approx1.15

Answer:

There is not enough evidence at the \(\alpha = 0.05\) level to support the researcher's concern that the percentage of high - school seniors who are smoking/ vaping nicotine is higher.