Question

a random sample of 110 professional athletes is chosen, and 13 score a ranking of \light speed\ while the makers say that their device is built to give most users room to improve. the makers say that even among professional athletes, the proportion p, who can score the top ranking of \light speed\ is less than 18%. (a) state the null hypothesis h0 and the alternative hypothesis h1 that you would use for the test. h0: p≥0.18, h1: p<0.18. (b) for your hypothesis test, you will use a z - test. find the values of np and n(1 - p) to confirm that a z - test can be used. (one standard is that np≥10 and n(1 - p)≥10 under the assumption that the null hypothesis is true.) here n is the sample size and p is the population proportion you are testing. np = 19.8, n(1 - p)=90.2. (c) perform a z - test and find the p - value. here is some information to help you with your z - test. the value of the test statistic is given by z = (p̂-p)/√(p(1 - p)/n). the p - value is the area under the curve to the left of the value of the test statistic.

Explanation:

Step1: Recall null and alternative hypotheses

The null hypothesis $H_0$ is a statement of no - effect or status - quo, and the alternative hypothesis $H_1$ is what we are trying to find evidence for. Given $H_0:p\geq0.18$ and $H_1:p < 0.18$, this is a left - tailed test.

Step2: Check the conditions for Z - test

We are given $np = 19.8$ and $n(1 - p)=90.2$. Since $np\geq10$ and $n(1 - p)\geq10$, the conditions for using a Z - test for a proportion are met. The sample size $n=19.8 + 90.2=110$.

Step3: Calculate the test statistic formula

The test statistic for a one - sample proportion Z - test is $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}$, where $\hat{p}$ is the sample proportion, $p$ is the hypothesized population proportion, and $n$ is the sample size. But we are not asked to calculate the test statistic value in full here.

Step4: Find the p - value

The p - value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true. For a left - tailed test with test statistic $z$, the p - value is $P(Z

Answer:

The null hypothesis $H_0:p\geq0.18$ and alternative hypothesis $H_1:p < 0.18$ is a left - tailed test. The conditions $np = 19.8\geq10$ and $n(1 - p)=90.2\geq10$ are met for using a Z - test. The p - value is the area under the standard normal curve to the left of the calculated test statistic $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}$.