Question 6: A manufacturer knows that the number of items produced per hour by its two factories A and B is normally distributed with standard deviations 8.0 and 11.0 items respectively. The mean hourly amount produced by Firm A from a random sample of 50 hours is 120 units and that by Firm B from a random sample of 30 hours is 110 units. Find the 95% confidence interval for the difference in the means and comment on the estimated interval.
You will need to use a two-sample confidence interval formula.
Here is one example:
CI95 = (x1 − x2) ± z-value × √(s1^2/n1 + s2^2/n2)
(Note: × means to multiply)
x1 = mean A
x2 = mean B
z-value = 1.96
s1^2 = standard deviation A (squared)
s2^2 = standard deviation B (squared)
n1 = sample size A
n2 = sample size B
Plug the values into the formula and calculate the interval.
I hope this will help get you started.
To find the 95% confidence interval for the difference in the means, we can follow these steps:
Step 1: Calculate the standard error of the difference in means.
First, we calculate the standard error for each sample.
For Firm A, the standard error (SE_A) can be calculated using the formula:
SE_A = standard deviation / √(sample size)
Given that the standard deviation for Firm A is 8.0 and the sample size is 50, we have:
SE_A = 8.0 / √50 ≈ 1.131
Similarly, for Firm B, the standard error (SE_B) can be calculated using the formula:
SE_B = standard deviation / √(sample size)
Given that the standard deviation for Firm B is 11.0 and the sample size is 30, we have:
SE_B = 11.0 / √30 ≈ 2.008
Step 2: Calculate the margin of error.
The margin of error (MOE) can be calculated by multiplying the standard error by the critical value from the t-distribution table. Since we want a 95% confidence interval, the critical value would correspond to an alpha value of 0.05 (5% significance level). For a two-tailed test, the critical value is obtained by dividing the alpha value by 2 and looking up the corresponding t-value for the degrees of freedom, which is the sum of the sample sizes minus 2 (50 + 30 - 2 = 78). From the t-distribution table, the critical value for alpha/2 = 0.025 and 78 degrees of freedom is approximately 1.990.
Therefore, the margin of error can be calculated as follows:
MOE = critical value * standard error
MOE = 1.990 * √((SE_A)^2 + (SE_B)^2)
MOE = 1.990 * √(1.131^2 + 2.008^2)
MOE ≈ 3.991
Step 3: Calculate the confidence interval.
Finally, we can calculate the confidence interval by subtracting and adding the margin of error from the difference in sample means.
Difference in sample means = mean of Firm A - mean of Firm B
Difference = 120 - 110 = 10
Confidence interval = (Difference - MOE, Difference + MOE)
Confidence interval = (10 - 3.991, 10 + 3.991)
Confidence interval ≈ (6.009, 13.991)
The 95% confidence interval for the difference in the means is approximately (6.009, 13.991).
Commentary on the estimated interval:
This means that, with 95% confidence, the true difference in the means of the two factories lies within this interval. Since the interval does not contain zero (0), we can conclude that the mean hourly amount produced by Firm A is statistically significantly different from Firm B at the 5% level of significance. The interval suggests that Firm A produces, on average, between 6.009 and 13.991 more units per hour than Firm B.