For one of the tasks in a manufacturing process, the mean time for task completion has historically been 35.0 minutes, with a standard deviation of 2.5 minutes. Workers have recently complained that the machinery used in the task is wearing out and slowing down. In response to the complaints, plant engineers have measured the time required for a sample consisting of 100 task operations. The 100 sample times, in minutes, re in data file XR09022. Using the mean for this sample, and assuming that the population standard deviation has remained unchanged at 2.5 minutes, construct the 95% confidence interval for the population mean. Is 35.0 minutes within the confidence interval? Interpret your “yes” or “no” answer in terms of whether the mean time for the task may have changed.
Formula:
CI95 = mean + or - 1.96(sd divided by √n)
...where + or - 1.96 represents the 95% confidence interval using a z-table, sd = standard deviation, √ = square root, and n = sample size.
With your data:
CI95 = mean + or - 1.96 (2.5/√100)
Note: Determine the mean for the sample to be used in the above calculation.
Finish the calculation and determine the interval. Determine if the 35 minutes is contained within the interval.
I hope this will help get you started.
To construct the 95% confidence interval for the population mean, we can use the formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
First, let's calculate the Sample Mean:
Sample Mean = Σx / n
where Σx is the sum of all the sample times and n is the sample size (100).
Next, let's calculate the Standard Error:
Standard Error = σ / √n
where σ is the population standard deviation (2.5 minutes) and √n is the square root of the sample size (square root of 100).
Finally, we need to determine the Critical Value for a 95% confidence level. Since the sample size is large (n > 30), we can use the Z-distribution and find the corresponding z-value for a 95% confidence level (z*).
With these calculations, we can construct the confidence interval by plugging in the values:
Confidence Interval = Sample Mean ± (z* * Standard Error)
If the confidence interval includes the mean time of 35.0 minutes, it suggests that the mean time for the task may not have changed significantly. If the confidence interval does not include 35.0 minutes, it suggests that the mean time for the task may have changed.
Now, let's proceed with the calculations.
To construct a confidence interval for the population mean, we can use the formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
1. Calculate the sample mean (x̄): Add up all the sample times and divide it by the total number of samples (100).
2. Calculate the standard error (SE): Divide the standard deviation by the square root of the sample size.
SE = Standard Deviation / √(Sample Size)
3. Find the critical value: Since we want to construct a 95% confidence interval, the level of confidence is 0.95. Convert this into a z-score using the standard normal distribution table. The critical value for a 95% confidence level is approximately 1.96.
4. Calculate the confidence interval: Plug the values calculated above into the formula.
Confidence Interval = x̄ ± (1.96 * SE)
5. Determine if the population mean of 35.0 minutes falls within the confidence interval. If the interval includes 35.0 minutes, we can say the mean time for the task may not have changed significantly. Otherwise, if 35.0 minutes is outside the confidence interval, we can suggest that the mean time might have changed.
By performing these calculations and considering the given information, the mean time for the task can be assessed.
Note: We need access to the data file XR09022 to calculate the sample mean and standard deviation.