To construct a 90% confidence interval for [tex]\(\mu_d\)[/tex], the population mean difference in assembly times for the two processes, we proceed with the following steps:
1. Determine the differences:
From the given data:
Differences (Process 1 - Process 2): [tex]\( [14, -2, -10, 18, -7, 14, 10, -12, 9, -9] \)[/tex]
2. Calculate the sample mean ([tex]\(\bar{x}\)[/tex]) of the differences:
The mean difference is given as:
[tex]\[
\bar{x} = 2.5
\][/tex]
3. Calculate the sample standard deviation ([tex]\(s\)[/tex]) of the differences:
The standard deviation of the differences is:
[tex]\[
s = 11.607
\][/tex]
4. Determine the number of samples (n):
The number of differences is:
[tex]\[
n = 10
\][/tex]
5. Determine the degrees of freedom (df):
[tex]\[
\text{df} = n - 1 = 10 - 1 = 9
\][/tex]
6. Determine the critical t-value for the 90% confidence level:
Using the t-distribution table or an appropriate statistical tool, we find the critical t-value for a 90% confidence level and 9 degrees of freedom:
[tex]\[
t_{\text{critical}} = 1.833
\][/tex]
7. Calculate the margin of error (ME):
The margin of error is given by:
[tex]\[
\text{ME} = t_{\text{critical}} \times \left(\frac{s}{\sqrt{n}}\right)
\][/tex]
Substituting the values, we get:
[tex]\[
\text{ME} = 1.833 \times \left(\frac{11.607}{\sqrt{10}}\right) = 6.728
\][/tex]
8. Calculate the lower and upper limits of the confidence interval:
The lower limit is:
[tex]\[
\text{Lower limit} = \bar{x} - \text{ME} = 2.5 - 6.728 = -4.23
\][/tex]
The upper limit is:
[tex]\[
\text{Upper limit} = \bar{x} + \text{ME} = 2.5 + 6.728 = 9.23
\][/tex]
Therefore, the 90% confidence interval for the population mean difference in assembly times for the two processes is:
- Lower limit: [tex]\(-4.23\)[/tex]
- Upper limit: [tex]\(9.23\)[/tex]
