A manufacturing company wishes to estimate the number of items that its workforce can produce on average each hour now that they have a new machine. The factory examined the records for a random sample of 8 hours over the past month. The hourly production rates for these 8 hours were:
142 175 162 158 190 154 160 185
(a) Calculate and interpret the 99 percent confidence interval for the average number of items produced per hour.
(b) What assumption did you make in order to answer part (a)?
To calculate the 99 percent confidence interval for the average number of items produced per hour, we can use the following steps:
Step 1: Calculate the sample mean (x̄)
The sample mean (x̄) can be calculated by summing up all the hourly production rates and dividing it by the total number of hours in the sample. In this case, the sum of the 8 production rates is:
142 + 175 + 162 + 158 + 190 + 154 + 160 + 185 = 1,326
So, the sample mean (x̄) is:
x̄ = 1,326 / 8 = 165.75
Step 2: Calculate the sample standard deviation (s)
The sample standard deviation (s) can be calculated using the following formula:
s = sqrt((∑(x - x̄)^2) / (n - 1))
where x is the production rate for each hour, x̄ is the sample mean, and n is the number of hours in the sample.
Substituting the given values into the formula:
s = sqrt((∑(142 - 165.75)^2 + (175 - 165.75)^2 + (162 - 165.75)^2 + (158 - 165.75)^2 + (190 - 165.75)^2 + (154 - 165.75)^2 + (160 - 165.75)^2 + (185 - 165.75)^2) / (8 - 1))
s = sqrt((3605 / 7)) = sqrt(515) ≈ 22.7
Step 3: Calculate the standard error (SE)
The standard error (SE) can be calculated by dividing the sample standard deviation (s) by the square root of the sample size (n):
SE = s / sqrt(n)
In this case, since the sample size is 8:
SE ≈ 22.7 / sqrt(8) ≈ 8.03
Step 4: Calculate the margin of error (ME)
The margin of error (ME) is calculated by multiplying the critical value (z) for the desired confidence level by the standard error (SE). For a 99 percent confidence level, the critical value is approximately 2.787.
ME = z * SE = 2.787 * 8.03 ≈ 22.37
Step 5: Calculate the confidence interval (CI)
The confidence interval (CI) is calculated by subtracting the margin of error (ME) from the sample mean (x̄) and adding it to the sample mean (x̄):
CI = x̄ - ME to x̄ + ME
In this case:
CI = 165.75 - 22.37 to 165.75 + 22.37
CI ≈ 143.38 to 188.12
(a) The 99 percent confidence interval for the average number of items produced per hour is approximately 143.38 to 188.12. This means that we are 99 percent confident that the true average number of items produced per hour falls within this interval.
(b) In order to calculate this confidence interval, we made the assumption that the distribution of the production rates is approximately normal. This assumes that the sample is random and independent, and that the population of production rates follows a normal distribution.