18 workers at a large assembly plant were asked the time it took to commute to work. The data was analyzed in Minitab and the Descriptive Statistics for the data appears below.
Descriptive Statistics: Minutes
Variable N Mean Median TrMean StDev SE Mean
Minutes 18 30.06 29.50 30.06 5.68 1.34
Variable Minimum Maximum Q1 Q3
Minutes 19.00 41.00 26.50 34.00
a) It is known that the standard deviation for all commute times is 6 minutes.
Construct a 98% confidence interval for the average commute time for all workers at the assembly plant. You may use any of the Minitab Output above.
b) Write a complete confidence statement for the interval you calculated in part a.
a) To construct a 98% confidence interval for the average commute time for all workers at the assembly plant, we can use the information provided in the Minitab output.
From the output, we see that the mean commute time (µ) is 30.06 minutes. The standard deviation (σ) for all commute times is given as 6 minutes. The sample size (N) is 18.
To calculate the confidence interval, we can use the formula:
Confidence interval = Mean ± (t-value * standard error)
The t-value depends on the confidence level and the sample size. In this case, we want a 98% confidence level and the sample size is 18. We can find the appropriate t-value from the t-distribution table or use Minitab.
Using the Minitab output, we see that the standard error (SE Mean) is 1.34 minutes.
To find the t-value, we need to determine the degrees of freedom (df). For a sample size of 18, the degrees of freedom is given by (N - 1) = 17.
Using the t-distribution table or Minitab, we find that the t-value for a 98% confidence level and 17 degrees of freedom is approximately 2.898.
Now we can calculate the confidence interval:
Confidence interval = 30.06 ± (2.898 * 1.34)
Confidence interval = 30.06 ± 3.888
Confidence interval ≈ (26.172, 33.948)
Therefore, the 98% confidence interval for the average commute time for all workers at the assembly plant is approximately (26.172 minutes, 33.948 minutes).
b) The complete confidence statement for the interval calculated in part a would be:
We are 98% confident that the true average commute time for all workers at the assembly plant lies between 26.172 minutes and 33.948 minutes.