Photon is a training device that is designed to improve a user's 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 120 professional athletes is chosen, and 14 score a ranking of "light speed" while using the device.

Question

Photon is a training device that is designed to improve a user's 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 120 professional athletes is chosen, and 14 score a ranking of "light speed" while using the device.
Complete the parts below to perform a hypothesis test to see if there is enough evidence, at the 0.10 level of significance, to support the claim that the proportion of all professional athletes who can score a ranking of "light speed" is less than 18%.

(a)	State the null hypothesis Hv0 and the alternative hypothesis Hv1 that you would use for the test.

GPT-4o mini · Answer

To conduct the hypothesis test regarding the proportion of professional athletes who can score a ranking of "light speed," we define the hypotheses as follows:

Null Hypothesis (H0):

The null hypothesis states that the proportion of all professional athletes who can score a ranking of "light speed" is equal to or greater than 18%. This can be expressed mathematically as: \[ H_0: p \geq 0.18 \]

Alternative Hypothesis (H1):

The alternative hypothesis states that the proportion of all professional athletes who can score a ranking of "light speed" is less than 18%. This can be expressed mathematically as: \[ H_1: p < 0.18 \]

In summary:

$ H_0: p \geq 0.18 $
$ H_1: p < 0.18 $

Answer

(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 is less than or equal to 10 and n(1-p) is less than or equal to 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) = [?]

GPT-4o mini · Answer

To use a Z-test for a hypothesis test regarding proportions, we need to check the conditions for normal approximation. Specifically, we will calculate $ np $ and $ n(1-p) $ using the values from the null hypothesis.

Sample Size (n): The sample size is given as $ n = 120 $.
Proposed Population Proportion (p): Under the null hypothesis, we will test using $ p = 0.18 $.

Now we can calculate $ np $ and $ n(1-p) $:

Calculations:

Calculate $ np $: \[ np = n \cdot p = 120 \cdot 0.18 = 21.6 \]
Calculate $ n(1-p) $: \[ n(1-p) = n \cdot (1 - p) = 120 \cdot (1 - 0.18) = 120 \cdot 0.82 = 98.4 \]

Summary of Results:

$ np = 21.6 $
$ n(1-p) = 98.4 $

Since $ np $ is greater than 10, the Z-test assumption is not fully met (the standard criterion requires $ np \leq 10 $). However, in practical scenarios, many researchers consider $ np $ and $ n(1-p) $ to still permit the use of the Z-test when both values are sufficiently large, instead of strictly adhering to "less than or equal to 10."

Thus, while technically this would not satisfy the strict criteria for using the Z-test, the values can still be used for hypothesis testing in practice. In strict accordance with the typical guidelines, the Z-test may not be appropriate here because $ np > 10 $.

If you need to adhere strictly to the criteria or explore alternative methods, a different approach like exact binomial tests might be warranted given the observed conditions.

Answer

this is one or two tailed? what is the test statistic (rounded to 3 decimal places?) what is the p-value (rounded to 3 decimal places?)

GPT-4o mini · Answer

### 1. One-tailed or Two-tailed Test: In this case, we are testing whether the proportion of professional athletes who can score a ranking of "light speed" is **less than** 18%. Because the alternative hypothesis $ H_1: p < 0.18 $ specifies a direction (less than), this is a **one-tailed test**. ### 2. Test Statistic: To calculate the test statistic for a proportion Z-test, we use the formula: $$ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} $$ Where: - $\hat{p} = \frac{x}{n}$ (the sample proportion) - $x$ = number of successes (in this case, athletes who scored "light speed") - $n$ = sample size - $p_0 = 0.18$ (the proportion under the null hypothesis) From your data: - $x = 14$ - $n = 120$ **Calculate $\hat{p}$**: $$ \hat{p} = \frac{14}{120} = 0.1167 $$ **Plug into the Z-test formula**: $$ Z = \frac{0.1167 - 0.18}{\sqrt{\frac{0.18(1 - 0.18)}{120}}} $$ $$ = \frac{0.1167 - 0.18}{\sqrt{\frac{0.18 \cdot 0.82}{120}}} $$ $$ = \frac{0.1167 - 0.18}{\sqrt{\frac{0.1476}{120}}} $$ $$ = \frac{0.1167 - 0.18}{\sqrt{0.001230}} $$ $$ = \frac{-0.0633}{0.0351} \approx -1.803 $$ **Rounded to three decimal places, the test statistic is**: $$ Z \approx -1.803 $$ ### 3. P-value: To find the p-value for the test statistic $ Z = -1.803 $, we can look it up in the standard normal distribution table or use a calculator. Using a Z-table or calculator: - The p-value corresponds to the area to the left of $ Z $ since this is a one-tailed test. For $ Z = -1.803 $: - The p-value is approximately $ 0.0356 $. **Rounded to three decimal places, the p-value is**: $$ \text{p-value} \approx 0.036 $$ ### Summary: - **Test Statistic (Z)**: -1.803 - **P-value**: 0.036 If you compare this p-value with your significance level (0.10), you can make a decision on the null hypothesis. Since $ 0.036 < 0.10 $, you would reject the null hypothesis in favor of the alternative.

Answer

(d)Based on your answer to part (c), choose what can be concluded, at the 0.10 level of significance, about the claim made by the makers of the device.

1) Since the p-value is less than (or equal to) the level of significance, the null hypothesis is rejected. So, there is enough evidence to support the claim that less than 18% of professional athletes can score the top-ranking of "light speed."

2) Since the p-value is less than (or equal to) the level of significance, the null hypothesis is not rejected. So, there is not enough evidence to support the claim that less than 18% of professional athletes can score the top-ranking of "light speed."

3) Since the p-value is greater than the level of significance, the null hypothesis is rejected. So, there is enough evidence to support the claim that less than 18% of professional athletes can score the top-ranking of "light speed."

or 4) Since the p-value is greater than the level of significance, the null hypothesis is not rejected. So, there is not enough evidence to support the claim that less than 18% of professional athletes can score the top-ranking of "light speed."

GPT-4o mini · Answer

Based on the results from part (c), where we found the p-value to be approximately 0.036, and comparing it with the significance level of 0.10, we can make the following conclusion:

Since the p-value (0.036) is less than the level of significance (0.10), we reject the null hypothesis. This indicates that there is enough evidence to support the claim that less than 18% of professional athletes can score the top-ranking of "light speed."

Thus, the correct conclusion is:

1) Since the p-value is less than (or equal to) the level of significance, the null hypothesis is rejected. So, there is enough evidence to support the claim that less than 18% of professional athletes can score the top-ranking of "light speed."