A random sample of 250 men yielded 175 who said they'd ridden a motorcycle at some time in their lives, while a similar sample of 215 women yielded only 43 that had done so. Find a 99% confidence interval for the difference between the proportions of men and women who have ridden motorcycles.
.4688 ± .085
.5 ± .112
.5 ± .103
.5 ± .085
.5 ± .078
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To find the 99% confidence interval for the difference between the proportions of men and women who have ridden motorcycles, we can use the formula:
CI = (p1 - p2) ± Z * √((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2))
Where:
p1 = proportion of men who have ridden motorcycles = 175/250 = 0.7
p2 = proportion of women who have ridden motorcycles = 43/215 = 0.2
n1 = sample size of men = 250
n2 = sample size of women = 215
Z = Z-score for a 99% confidence level, which is approximately 2.58
Plugging in the values:
CI = (0.7 - 0.2) ± 2.58 * √((0.7 * (1 - 0.7) / 250) + (0.2 * (1 - 0.2) / 215))
CI = 0.5 ± 2.58 * √((0.7 * 0.3 / 250) + (0.2 * 0.8 / 215))
CI = 0.5 ± 2.58 * √(0.00126 + 0.00149)
CI = 0.5 ± 2.58 * √(0.00275)
CI = 0.5 ± 2.58 * 0.05244
CI = 0.5 ± 0.135
Therefore, the 99% confidence interval for the difference between the proportions of men and women who have ridden motorcycles is:
0.5 ± 0.135
The correct answer is either option:
.5 ± .112
or
.5 ± .103
To find a 99% confidence interval for the difference between the proportions of men and women who have ridden motorcycles, we can use the formula:
Confidence Interval = (p1 - p2) ± Z * sqrt((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2))
where p1 and p2 are the sample proportions, n1 and n2 are the sample sizes, and Z is the Z-score for the desired confidence level.
Here, for men, out of 250 men, 175 said they've ridden a motorcycle, so the sample proportion is p1 = 175/250 = 0.7. For women, out of 215 women, 43 said they've ridden a motorcycle, so the sample proportion is p2 = 43/215 = 0.2.
Substituting the values into the formula, and using a Z-score of 2.58 (for a 99% confidence level), we get:
Confidence Interval = (0.7 - 0.2) ± 2.58 * sqrt((0.7 * (1 - 0.7) / 250) + (0.2 * (1 - 0.2) / 215))
Calculating the values within the square root:
((0.7 * 0.3) / 250) + ((0.2 * 0.8) / 215) ≈ 0.006
Then substituting this value into the formula:
Confidence Interval = 0.5 ± 2.58 * sqrt(0.006)
Calculating the square root of 0.006:
sqrt(0.006) ≈ 0.077
Finally, substituting this value into the confidence interval:
Confidence Interval = 0.5 ± 2.58 * 0.077
Calculating:
Confidence Interval = 0.5 ± 0.199
Therefore, the correct answer is .5 ± .199.
So, the correct option from the given choices is .5 ± .199.