Question

photon is a training device that is designed to improve a users reaction time. similar devices have been criticized for being too easy to master, but the makers of photon 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 random sample of 110 professional athletes is chosen, and 13 score a ranking of \light speed\. (a) state the null hypothesis h0 and the alternative hypothesis h1 that you would use for the test. h0:□ h1:□ (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 = □ n(1 - p)=□ (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. standard normal distribution step 1: select one - tailed or two - tailed. o one - tailed

Explanation:

Step1: Set up null and alternative hypotheses

The claim is that the proportion \(p\) of professional athletes who score a ranking of "light - speed" is less than \(0.18\). So, the null hypothesis \(H_0\) is the statement of no effect, and the alternative hypothesis \(H_1\) is the claim we are trying to find evidence for.
\(H_0:p\geq0.18\) and \(H_1:p < 0.18\)

Step2: Calculate \(np\) and \(n(1 - p)\)

We are given \(n = 110\) and \(p=0.18\).
\(np=110\times0.18 = 19.8\)
\(n(1 - p)=110\times(1 - 0.18)=110\times0.82 = 90.2\)

Step3: Perform the Z - test

The test - statistic for a proportion is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\), but we are not asked to calculate the test - statistic in full here. The p - value is the area under the standard normal curve to the left of the test statistic for a left - tailed test.

Answer:

(a) \(H_0:p\geq0.18\), \(H_1:p < 0.18\)
(b) \(np = 19.8\), \(n(1 - p)=90.2\)