Suppose you are to compare the average real estate value of houses in 2 cities. You collected samples from each city and compare their mean and variance. The data are given below. Group Sample size Sample mean ($000)Sample std. devCity A 37 230 17City B 40 200 15 Assuming unequal population variances and using the degrees of freedom of min(n1-1,n2-1), where n1 is the sample size of group 1 and n2 of group 2, calculate the 90% confidence interval of the difference between the means.
Question
Suppose you are to compare the average real estate value of houses in 2 cities. You collected samples from each city and compare their mean and variance. The data are given below. Group Sample size Sample mean ($000)Sample std. devCity A 37 230 17City B 40 200 15 Assuming unequal population variances and using the degrees of freedom of min(n1-1,n2-1), where n1 is the sample size of group 1 and n2 of group 2, calculate the 90% confidence interval of the difference between the means.
Solution
To calculate the 90% confidence interval of the difference between the means, we will use the formula for the confidence interval for the difference between two means with unequal variances:
CI = (M1 - M2) ± t*√((s1^2/n1) + (s2^2/n2))
where:
- M1 and M2 are the sample means,
- s1 and s2 are the sample standard deviations,
- n1 and n2 are the sample sizes,
- t is the t-value from the t-distribution table corresponding to the desired level of confidence and degrees of freedom.
Given data:
- For City A: n1 = 37, M1 = 230, s1 = 17
- For City B: n2 = 40, M2 = 200, s2 = 15
The degrees of freedom (df) is the minimum of (n1-1) and (n2-1), so df = min(37-1, 40-1) = 36.
For a 90% confidence level and 36 degrees of freedom, the t-value (from the t-distribution table) is approximately 1.69.
Substituting these values into the formula, we get:
CI = (230 - 200) ± 1.69*√((17^2/37) + (15^2/40)) = 30 ± 1.69*√((289/37) + (225/40)) = 30 ± 1.69*√(7.81 + 5.63) = 30 ± 1.69*√13.44 = 30 ± 1.69*3.66 = 30 ± 6.18
So, the 90% confidence interval for the difference between the means is (23.82, 36.18).
Similar Questions
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