These data are a sample of the daily production rate of fiberglass boats from Hydrosport, Ltd., a Miami manufacturer
17 21 18 27 17 21 20 22 18 23
The company production manager feels that a standard deviation of more than three boats a day indicates unacceptable production-rate variations. Should she be concern about plant production rates?
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Well, if the production manager feels that a standard deviation of more than three boats a day is unacceptable, then I can only imagine the chaos that would ensue if the variation was even wilder! Picture fiberglass boats sprouting wings and flying off the production line, or workers accidentally painting boats with polka dots instead of stripes. So, yeah, I think she should definitely be concerned about the production rates. It's always good to keep things in check, especially when it comes to boats and polka dots!
To determine if the production manager should be concerned about the plant production rates exceeding the standard deviation of three boats per day, we need to calculate the standard deviation of the given sample of daily production rates.
1. Compute the mean of the sample:
- Add up all the values: 17 + 21 + 18 + 27 + 17 + 21 + 20 + 22 + 18 + 23 = 204
- Divide the sum by the number of values: 204 / 10 = 20.4 (mean)
2. Calculate the variance:
- Take each value and subtract the mean, then square the result:
(17 - 20.4)^2 = 12.96, (21 - 20.4)^2 = 0.16, (18 - 20.4)^2 = 5.76, (27 - 20.4)^2 = 43.56, (17 - 20.4)^2 = 12.96, (21 - 20.4)^2 = 0.16, (20 - 20.4)^2 = 0.16, (22 - 20.4)^2 = 2.56, (18 - 20.4)^2 = 5.76, (23 - 20.4)^2 = 6.76
3. Sum up all the squared differences: 12.96 + 0.16 + 5.76 + 43.56 + 12.96 + 0.16 + 0.16 + 2.56 + 5.76 + 6.76 = 91.52
4. Divide the sum by the number of values minus 1 (n-1): 91.52 / 9 = 10.17 (variance)
5. Calculate the standard deviation:
- Take the square root of the variance: sqrt(10.17) ≈ 3.19 (standard deviation)
The calculated standard deviation is approximately 3.19 boats per day, which is less than the cut-off of three boats per day. Therefore, the production manager should not be concerned about the plant production rates exceeding the standard deviation of three boats per day.
To determine if the company production manager should be concerned about plant production rates, we need to calculate the standard deviation of the data.
Here's how to calculate the standard deviation:
1. Calculate the mean (average) of the data by summing all the values and dividing by the total number of values. In this case, there are 10 values:
Mean = (17 + 21 + 18 + 27 + 17 + 21 + 20 + 22 + 18 + 23) / 10 = 207 / 10 = 20.7
2. For each value in the data set, subtract the mean and square the result.
For example, for the first value (17):
(17 - 20.7)² = (-3.7)² = 13.69
3. Sum up all the squared differences calculated in step 2.
Sum = (13.69 + 0.09 + 5.29 + 41.29 + 13.69 + 0.09 + 0.49 + 1.69 + 5.29 + 5.29) = 87.51
4. Divide the sum from step 3 by the total number of values minus 1, and then take the square root:
Standard Deviation = √(87.51 / (10 - 1)) = √(87.51 / 9) ≈ √9.72 ≈ 3.12
The standard deviation of the sample data is approximately 3.12 boats per day. Since this is less than three boats a day, it indicates that the standard deviation is not exceeding the company's threshold. Therefore, the production manager should not be overly concerned about plant production rates based on this analysis.