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:

= \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.