Question

Monitoring weights of flour bags.When it is working properly, a machine that fills 25-pound bags of flour dispenses an average of 25 pounds per fill; the standard deviation of the amount of fill is .1 pound. To monitor the performance of the machine, an inspector weighs the contents of a bag coming off the machine’s conveyor belt every half hour during the day. If the contents of two consecutive bags fall more than 2 standard deviations from the mean (using the mean and standard deviation given above), the filling process is said to be out of control, and the machine is shut down briefly for adjustments. The data

given in the following table are the weights measured by the inspector yesterday. Assume the machine is never shut down for more than 15 minutes at a time. At what times yesterday was the process shut down for adjustment? Justify your answer.

Time	Weight (pounds)
8:00 a.m	25.10
8:30	25.15
9:00	24.81
9:30	24.75
10:00	25.00
10:30	25.05
11:00	25.23
11:30	25.25
12:00	25.01
12:30 p.m	25.06
1:00	24.95
1:30	24.80
2:00	24.95
2:30	25.21
3:00	24.90
3:30	24.71
4:00	25.31
4:30	25.15
5:00	25.20

Short answer

11:30 am and 4 pm

Step-by-step solution

Calculating the interval

The machine requires adjustment time if 2 bags of flour exceed more than 2 standard deviations from the mean consecutively. So we will first calculate that interval using $\overline{x}\pm 2s$

$$\begin{gathered} \overline{x}\pm 2s \\ Lowerrange=\overline{x}-2s \\ =25-2(0.1) \\ =25-0.2 \\ =24.8 \\ Upperrange=\overline{x}+2s \\ =25+2(0.1) \\ =25+0.2 \\ =25.2 \end{gathered}$$

Therefore, the 2 standard deviations from the mean interval is 24.8 pounds per bag to 25.2 pounds per bag.

Finding the number of observations that fall in this range

We find that 14 observations fall within 24.8 to 25.2 pounds per bag.

2 observations fall below 24.8, and 4 observations fall above 25.2 pounds per bag.

Estimating the time of adjustment

Out of the outlier observations above 25.2 pounds per bag and below 24.8, the two times when consecutive observations are beyond standard deviations from the mean are

1. 25.23 and 25.25 pounds per bag at 11 and 11:30 am, respectively.
2. 24.71 and 25.31 pounds per bag at 3:30 and 4 pm, respectively.

Therefore, the machine was shut off for adjustment at 11:30 am and 4 pm.