In a random sample of 63 women at a company, the mean salary is $48,902 with a standard deviation of $5270. In a random sample of 50 men at the company, the mean salary is $53,454 with a standard deviation of $4677. Construct a 95% confidence interval for the difference between the mean salaries of all women and men at the company. show work
($2134.05, $6969.95)
($2715.11, $6388.89)
($3614.81, $5489.19)
($4083.40, $5020.60)
To construct a 95% confidence interval for the difference between the mean salaries of all women and men at the company, we can use the formula:
Confidence interval = (mean difference) ± (critical value) * (standard error)
1. Calculate the mean difference:
The mean difference between the salaries of women and men is the difference between their sample means.
Mean difference = Mean salary of women - Mean salary of men
= $48,902 - $53,454
= -$4,552
2. Calculate the standard error:
The standard error measures the variability or uncertainty in the estimate of the mean difference. It is calculated using the formula:
Standard error = sqrt[(standard deviation of women)^2 / sample size of women + (standard deviation of men)^2 / sample size of men]
= sqrt[(5270^2 / 63) + (4677^2 / 50)]
= sqrt[(27772900 / 63) + (21871209 / 50)]
= sqrt(440925.4 + 437424.18)
= sqrt(878349.58)
≈ 936.66
3. Determine the critical value:
To construct a 95% confidence interval, we need to find the critical value corresponding to the level of confidence. Since we want a 95% confidence interval, the critical value can be found using a standard normal distribution table or a calculator.
The critical value for a 95% confidence interval is approximately 1.96.
4. Calculate the confidence interval:
Using the formula for the confidence interval mentioned earlier:
Confidence interval = (mean difference) ± (critical value) * (standard error)
= (-4552) ± (1.96) * (936.66)
= (-4552) ± (1833.04)
So, the 95% confidence interval for the difference between the mean salaries of all women and men at the company is:
Confidence interval ≈ ($-6,385.04, $-719.96)
The answer options provided do not match the calculated confidence interval. Therefore, none of the given options seem to be correct. It appears there is an error in the provided answer choices.
To construct a 95% confidence interval for the difference between the mean salaries of all women and men at the company, we can use the formula:
CI = (x̄₁ - x̄₂) ± (t * SE)
Where:
x̄₁ = mean salary of women
x̄₂ = mean salary of men
t = critical value (obtained from t-distribution table for a desired confidence level and degrees of freedom)
SE = standard error of the difference
Calculating the standard error:
SE = sqrt[(σ₁² / n₁) + (σ₂² / n₂)]
Given:
n₁ = 63 (sample size for women)
n₂ = 50 (sample size for men)
σ₁ = 5270 (standard deviation for women)
σ₂ = 4677 (standard deviation for men)
SE = sqrt[(5270² / 63) + (4677² / 50)]
SE = sqrt[(27772900 / 63) + (21848129 / 50)]
SE = sqrt[441074.6032 + 436962.58]
SE = sqrt(877037.1832)
SE ≈ 936.42
Now, we need to find the critical value (t) for a 95% confidence level. For a two-tailed test and degrees of freedom (df) = n₁ + n₂ - 2 = 63 + 50 - 2 = 111, we find t = 1.984 (from the t-distribution table).
Substituting the values into the confidence interval formula:
CI = (48902 - 53454) ± (1.984 * 936.42)
CI = -4552 ± 1854.14
CI ≈ (-6406.14, -2697.86)
The correct answer is not among the provided options. However, the closest one is ($4083.40, $5020.60).