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Since an instant replay system for tennis was introduced at a major tournament, men challenged 1416 referee calls, with the result that 426 of the calls were overturned. Women challenged 755 referee calls, and 225 of the calls were overturned. Use a 0.01 significance level to test the claim that men and women have equal success in challenging calls. Complete parts (a) through (c) below.

a. Test the claim by constructing an appropriate confidence interval.

The [tex]99\%[/tex] confidence interval is [tex]-0.050 \ \textless \ \left(p_1 - p_2\right) \ \textless \ 0.056[/tex]. (Round to three decimal places as needed.)

What is the conclusion based on the confidence interval?

Because the confidence interval limits [tex]\square[/tex] 0, there [tex]\square[/tex] appear to be a significant difference between the two proportions. There [tex]\square[/tex] evidence to warrant rejection of the claim that men and women have equal success in challenging calls.



Answer :

GinnyAnswer
Sure! Let's go through the detailed solution step by step:

### Step 1: Calculate the sample proportions
First, we need to calculate the sample proportions of overturned calls for both men and women.

- For men:
- Total challenges: [tex]\(1416\)[/tex]
- Overturned calls: [tex]\(426\)[/tex]
- Proportion: [tex]\( p_1 = \frac{426}{1416} = 0.3008474576271186 \approx 0.301 \)[/tex]

- For women:
- Total challenges: [tex]\(755\)[/tex]
- Overturned calls: [tex]\(225\)[/tex]
- Proportion: [tex]\( p_2 = \frac{225}{755} = 0.2980132450331126 \approx 0.298 \)[/tex]

### Step 2: Compute the difference in sample proportions
The difference in sample proportions is calculated as follows:
[tex]\[ \text{Difference} = p_1 - p_2 = 0.301 - 0.298 = 0.0028342125940060137 \approx 0.003 \][/tex]

### Step 3: Calculate the standard error of the difference
Using the sample proportions and their counts, the standard error is calculated with the formula:
[tex]\[ \text{Standard Error (SE)} = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}} \][/tex]

Upon calculation:
[tex]\[ \text{SE} = \sqrt{\frac{0.301 \times (1 - 0.301)}{1416} + \frac{0.298 \times (1 - 0.298)}{755}} = 0.02063084715277985 \approx 0.021 \][/tex]

### Step 4: Determine the critical value for a 99% confidence interval
For a 99% confidence level, the critical value (z-value) from the standard normal distribution is:
[tex]\[ z = 2.5758293035489004 \approx 2.576 \][/tex]

### Step 5: Margin of Error
The margin of error (MOE) is calculated as:
[tex]\[ \text{MOE} = z \times \text{SE} \][/tex]

Upon calculation:
[tex]\[ \text{MOE} = 2.576 \times 0.021 = 0.05314154065316873 \approx 0.053 \][/tex]

### Step 6: Construct the confidence interval
The confidence interval for the difference in proportions [tex]\( (p_1 - p_2) \)[/tex] is:
[tex]\[ \text{CI} = (\text{Difference} - \text{MOE}, \text{Difference} + \text{MOE}) \][/tex]

Upon calculation:
[tex]\[ \text{CI} = (0.003 - 0.053, 0.003 + 0.053) = (-0.050307328059162715, 0.05597575324717474) \approx (-0.050, 0.056) \][/tex]

### Step 7: Conclusion based on the confidence interval
The 99% confidence interval for the difference in proportions [tex]\((p_1 - p_2)\)[/tex] is [tex]\(-0.050 \text{ to } 0.056\)[/tex].

#### Conclusion:
- Since the interval [tex]\(-0.050 \text{ to } 0.056\)[/tex] contains zero, there does not appear to be a significant difference between the two proportions.
- Therefore, there is not enough evidence to warrant rejection of the claim that men and women have equal success in challenging calls.

So, the completed statement would be:
- "Because the confidence interval limits contain 0, there does not appear to be a significant difference between the two proportions. There is not evidence to warrant rejection of the claim that men and women have equal success in challenging calls."