One operation of a mill is to cut pieces of steel into parts that will later be used as the frame for front seats in an automobile. The steel is cut with a diamond saw and requires

the resulting parts to be within ±0.005 inch of the length specified by the automobile company. The data in the file
comes from a sample of 100 steel parts. The measurement reported is the difference in inches between the actual
length of the steel part, as measured by a laser measurement device, and the specified length of the steel part. For example, the first value, 0.002, represents a steel part that is 0.002 inch shorter than the specified length.
a. Construct a percentage histogram.
b. Is the steel mill doing a good job in meeting the requirements set by the automobile company? Explain.

Steel File:

Error
-0.002
0.0005
0.0025
0.001
0.002
0.001
0.005
-0.002
0
0.001
-0.0025
-0.003
0.001
-0.0005
0
-0.003
-0.001
0.0005
0.0025
-0.0025
0.002
0.001
0.001
0.001
-0.002
-0.003
-0.0015
-0.0005
0
-0.0025
-0.003
-0.001
0.002
-0.001
0
0.003
0.0015
0
0
-0.0025
0.0005
0.001
0.0005
0.001
0.0025
0.001
0
-0.0025
-0.001
0.0035
0
-0.003
-0.002
-0.001
-0.0015
-0.0015
-0.002
0.001
-0.0025
-0.0005
-0.0015
-0.0005
-0.0015
-0.0005
-0.0005
-0.001
-0.0015
0.0025
0.001
0.0005
0
-0.002
-0.0005
0.0005
0.0005
0
-0.0005
0.002
0.001
-0.0015
-0.001
-0.001
0.0005
-0.0025
0.0025
0.002
0.0025
-0.002
0
-0.0005
-0.001
0
-0.002
0
0
0.002
-0.003
-0.0005
-0.002
-0.002

-0.001

-0.001
0.001
0.001
0.001
0.0005
0.0005
0.001
-0.001
-0.001
-0.0005
-0.001
-0.0005
0.0005
0.001
0.0005
0.0005
0.001
0.0005
0.0005
0.0005

a. A percentage histogram of the data can be constructed by counting the number of values that fall within each range of errors and then converting that count into a percentage of the total number of values. For example, the range of errors from -0.003 to -0.002 includes 4 values, which is 4% of the total number of values.

b. The steel mill is doing a good job in meeting the requirements set by the automobile company. The data shows that the majority of the steel parts are within ±0.005 inch of the specified length, with the majority of the errors being less than 0.002 inch. This indicates that the steel mill is meeting the requirements set by the automobile company.

To construct a percentage histogram, we first need to calculate the frequencies of the different error values. This can be done by counting how many times each error value appears in the given file.

Let's calculate the frequencies:

-0.002: 7
0.0005: 5
0.0025: 6
0.001: 13
0.005: 1
0: 8
-0.0025: 8
-0.003: 5
-0.0005: 10
-0.0015: 6
0.003: 3
0.0015: 2
0.002: 10
0.0035: 1

Now, let's calculate the percentages of each frequency. We divide each frequency by the total number of steel parts (100) and multiply by 100 to get the percentage:

-0.002: 7/100 * 100 = 7%
0.0005: 5/100 * 100 = 5%
0.0025: 6/100 * 100 = 6%
0.001: 13/100 * 100 = 13%
0.005: 1/100 * 100 = 1%
0: 8/100 * 100 = 8%
-0.0025: 8/100 * 100 = 8%
-0.003: 5/100 * 100 = 5%
-0.0005: 10/100 * 100 = 10%
-0.0015: 6/100 * 100 = 6%
0.003: 3/100 * 100 = 3%
0.0015: 2/100 * 100 = 2%
0.002: 10/100 * 100 = 10%
0.0035: 1/100 * 100 = 1%

Now, we can construct the percentage histogram. This histogram will have the error values on the x-axis and the corresponding percentages on the y-axis. We can use bars to represent each error value, with the height of each bar representing the percentage.

Here is the percentage histogram:
```
Error | Percentage
--------------------
-0.003 | 5%
-0.0025 | 8%
-0.002 | 10%
-0.0015 | 6%
-0.001 | 13%
-0.0005 | 10%
0 | 8%
0.0005 | 5%
0.001 | 13%
0.0015 | 2%
0.002 | 10%
0.0025 | 6%
0.003 | 3%
0.0035 | 1%
0.005 | 1%
```

Now, let's analyze whether the steel mill is doing a good job in meeting the requirements set by the automobile company. The requirement is that the resulting parts should be within ±0.005 inch of the specified length.

Looking at the percentage histogram, we can see that the majority of error values fall within ±0.005 inch. The percentages of the error values greater than ±0.005 inch are relatively low (e.g., 5%, 1%). This suggests that the steel mill is doing a relatively good job in meeting the requirements set by the automobile company.

However, it's important to note that a thorough analysis would also consider the specific tolerances set by the automobile company and compare the percentages with those tolerances. Additionally, further statistical analysis such as calculating the mean and standard deviation of the errors could provide a more comprehensive assessment of the steel mill's performance.

To construct a percentage histogram, follow these steps:

Step 1: Determine the range of the data. To do this, find the minimum and maximum values in the dataset. In this case, the minimum value is -0.003 inch and the maximum value is 0.005 inch.

Step 2: Determine the number of equal-width intervals for the histogram. One common rule is to use the square root of the number of data points for the number of intervals. Since we have 100 data points, we can use 10 intervals.

Step 3: Determine the width of each interval. To do this, subtract the minimum value from the maximum value and divide it by the number of intervals. In this case, the width would be (0.005 - (-0.003)) / 10 = 0.0008 inch.

Step 4: Create a table to record the frequency of each interval. Label the intervals from the minimum value to the maximum value, incrementing by the width determined in step 3.

Interval | Frequency
------------------------------------|---------------
-0.003 to -0.0022 | 0
-0.0022 to -0.0014 | 0
-0.0014 to -0.0006 | 0
-0.0006 to 0.0002 | 0
0.0002 to 0.001 | 0
0.001 to 0.0018 | 0
0.0018 to 0.0026 | 0
0.0026 to 0.0034 | 0
0.0034 to 0.0042 | 0
0.0042 to 0.005 | 0

Step 5: Count the number of data points that fall into each interval and record it in the frequency column of the table.

For example, to count the number of data points in the first interval (-0.003 to -0.0022 inch), we would see that none of the data points fall within that range, so the frequency is zero.

Repeat this process for each interval.

Once you have filled in all the frequencies, you can construct the percentage histogram by dividing each frequency by the total number of data points (100), and then multiplying by 100 to get the percentage.

To determine if the mill is doing a good job in meeting the requirements set by the automobile company, we need to analyze the histogram. Specifically, we need to examine the frequencies in the intervals that fall within the ±0.005 inch requirement.

If the frequencies in these intervals are high, it indicates that a significant number of parts are within the specified length range, meaning the mill is doing a good job. Conversely, if the frequencies in these intervals are low, it indicates that a significant number of parts are outside the specified length range, meaning the mill is not meeting the requirement.

Based on the information provided, we do not have the histogram with frequencies, so we cannot determine whether the mill is doing a good job or not.