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Topic 5 Homework (Nonadaptive)
Question 13 of 21 (1 point)|Question Attempt: 1 of Unlimited

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Question 13
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 24%. 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 110 professional athletes use it; 22 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.

=z

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### Explanation:

To determine whether the claim made by the device makers is correct, we need to perform a hypothesis test. Specifically, we'll use a **Z-test** for proportions. Let's go step by step to solve the problem and find the value of the test statistic.

---

### Step 1: **Set Up Hypotheses**

- **Null Hypothesis (H₀):** The proportion of athletes who score the top ranking is \( p = 0.24 \) (24%).
- **Alternative Hypothesis (H₁):** The proportion of athletes who score the top ranking is \( p < 0.24 \) (this is a one-tailed test since we are checking if the proportion is *less than* 24%).

---

### Step 2: **Identify the Given Information**

- Sample size \( n = 110 \)
- Number of athletes who scored "light speed" \( x = 22 \)
- Sample proportion \( \hat{p} = \frac{x}{n} = \frac{22}{110} = 0.2 \)
- Population proportion \( p_0 = 0.24 \)

---

### Step 3: **Check Conditions for Z-Test**

The Z-test is appropriate if:
- The sample is random.
- The sample size is large enough for normal approximation (both \( n \cdot p_0 \) and \( n \cdot (1 - p_0) \) must be at least 5).

Let's check:
- \( n \cdot p_0 = 110 \cdot 0.24 = 26.4 \geq 5 \)
- \( n \cdot (1 - p_0) = 110 \cdot 0.76 = 83.6 \geq 5 \)

Since both conditions are satisfied, it is appropriate to use a Z-test.

---

### Step 4: **Calculate the Test Statistic (z)**

The formula for the Z-test statistic for proportions is:

\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}
\]

Substitute the values:

\[
z = \frac{0.2 - 0.24}{\sqrt{\frac{0.24 \cdot (1 - 0.24)}{110}}}
= \frac{-0.04}{\sqrt{\frac{0.24 \cdot 0.76}{110}}}
= \frac{-0.04}{\sqrt{\frac{0.1824}{110}}}
= \frac{-0.04}{\sqrt{0.0016582}}
= \frac{-0.04}{0.04071}
= -0.9835
\]

So, the value of the test statistic is:

\[
z \approx -0.984 \text{ (rounded to three decimal places)}
\]

---

### Conclusion:

The Z-test statistic is \( z \approx -0.984 \). You can use this value to find the p-value and make a decision about the null hypothesis.

---

Do you need details on how to interpret the test statistic, or would you like assistance with the next step (finding the p-value)?

---

### Follow-up Questions:
1. How do we interpret a Z-test statistic value in hypothesis testing?
2. What is the critical value for a one-tailed Z-test at a significance level of 0.05?
3. How do we compute the p-value from the Z-score?
4. What conclusion can be drawn if the p-value is smaller than the significance level?
5. How does increasing the sample size affect the test statistic?

### Tip:
Always verify that conditions for using the Z-test are met (e.g., large enough sample size) to ensure your hypothesis test is valid.

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 24%. 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 110 professional athletes use it; 22 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.

Learn how to perform a Z-test for proportions to test a claim about a reaction time device. The problem involves a sample of 110 professional athletes, with 22 achieving the top ranking. Key concepts include hypothesis testing, Z-test, and proportions. Suitable for Grades 11-12 and college students.

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