8.4 HWK: Hypothesis Tests for Proportions
Cuestion 3 of 8 (4 points) I Question Attempt: 1 of Unlimited
Britta
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A certain training device measures reaction times of users by illuminating lights, one at a time, and measuring the time it takes, the user to press each light to turn it off. The makers of the device are marketing it for high-level training, saying that even among professional athletes, the proportion who can score the top ranking of "light speed" is less than $23 \%$. As a fitness trainer who wants to buy the device to attract more customers, you want to feel comfortable that the claim made by the makers is correct. To test the claim, you decide to perform a hypothesis test. To do so, you rent the device and have a random sample of 125 professional athletes use it; 25 score a ranking of "light speed."

You confirm that it is appropriate to perform a Z-test.
Why is a $Z$-test appropriate?
Find $z$, the value of the test statistic for your $Z$-test. Round your answer to three or more decimal places.
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Step 1 ：A certain training device measures reaction times of users by illuminating lights, one at a time, and measuring the time it takes, the user to press each light to turn it off. The makers of the device are marketing it for high-level training, saying that even among professional athletes, the proportion who can score the top ranking of 'light speed' is less than $23 \%$. As a fitness trainer who wants to buy the device to attract more customers, you want to feel comfortable that the claim made by the makers is correct. To test the claim, you decide to perform a hypothesis test. To do so, you rent the device and have a random sample of 125 professional athletes use it; 25 score a ranking of 'light speed.'

Step 2 ：We need to calculate the Z-test statistic using the formula mentioned above. The sample proportion $\hat{p}$ is 25/125 = 0.2, the population proportion $p_0$ is 0.23, and the sample size $n$ is 125.

Step 3 ：Let's plug these values into the formula and calculate the Z-test statistic. The formula for the Z-test statistic is \(Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\).

Step 4 ：Substituting the given values into the formula, we get \(Z = \frac{0.2 - 0.23}{\sqrt{\frac{0.23(1 - 0.23)}{125}}}\).

Step 5 ：The calculated Z-test statistic is approximately -0.797. This value is negative, indicating that the sample proportion is less than the population proportion, which is consistent with the claim made by the makers of the device.

Step 6 ：Final Answer: The value of the test statistic for the Z-test is approximately \(\boxed{-0.797}\).