The foot lengths of a random sample of 63 men had a mean of 28 cm and a standard deviation of 2.8 cm, while a random sample of 90 women had a mean foot length of 24.5 cm and a standard deviation of 2.8 cm. Compute an approximate 95% confidence interval for the difference between the mean foot lengths of men and women. (Round all answers to the nearest tenth.)
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To compute the approximate 95% confidence interval for the difference between the mean foot lengths of men and women, we can use the formula:
Confidence Interval = (X_bar1 - X_bar2) ± Z * SE
Where:
X_bar1 = sample mean for men's foot length
X_bar2 = sample mean for women's foot length
Z = z-value corresponding to the desired level of confidence (95% confidence level has a z-value of approximately 1.96)
SE = standard error of the difference between the two sample means
First, let's calculate the standard error of the difference between the two sample means using the formula:
SE = sqrt((σ1^2 / n1) + (σ2^2 / n2))
Where:
σ1 = population standard deviation for men's foot length (given as 2.8 cm)
σ2 = population standard deviation for women's foot length (given as 2.8 cm)
n1 = sample size for men (given as 63)
n2 = sample size for women (given as 90)
SE = sqrt((2.8^2 / 63) + (2.8^2 / 90))
SE = sqrt((7.84 / 63) + (7.84 / 90))
SE = sqrt(0.124 / 0.711)
SE = sqrt(0.1747)
SE ≈ 0.418 cm (rounded to three decimal places)
Now, we can calculate the confidence interval using the formula:
Confidence Interval = (28 - 24.5) ± 1.96 * 0.418
Confidence Interval = 3.5 ± (1.96 * 0.418)
Confidence Interval = 3.5 ± 0.8208
Confidence Interval = (2.68, 4.32) cm (rounded to one decimal place)
Therefore, the approximate 95% confidence interval for the difference between the mean foot lengths of men and women is 2.7 cm to 4.3 cm.