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According to the journal *Chemical Engineering*, an important property of a fiber is its water absorbency. A random sample of 20 pieces of cotton fiber was taken and the absorbency on each piece was measured. The following are the absorbency values:

18.71, 21.41, 20.72, 21.81, 19.29, 22.43, 20.17,
23.71, 19.44, 20.50, 18.92, 20.33, 23.00, 22.85,
19.25, 21.77, 22.11, 19.77, 18.04, 21.12

Required:
a. Calculate the sample mean and median for the above sample values.
b. Compute the 10% trimmed mean.
c. Do a dot plot of the absorbency data.
d. Using only the values of the mean, median, and trimmed mean, do you have evidence of outliers in the data?

Answer :

a. The sample mean and median for the above sample values are approximately 20.98 and 20.61, respectively.

b. The 10% trimmed mean is approximately 20.70.

c. I'm unable to create a visual plot in this text-based format, but you can create a dot plot by plotting each data point as a dot along a number line.

d. Since the mean is slightly higher than the median and trimmed mean, there may be some positive outliers in the data, but the difference is not substantial.

a. To calculate the sample mean and median for the given data:

Sample Mean (Average):

To find the mean, sum up all the values and divide by the number of values (20):

Mean = (18.71 + 21.41 + 20.72 + ... + 21.12) / 20

Mean ≈ 20.98 (rounded to two decimal places)

Sample Median (Middle Value):

To find the median, first, order the data in ascending order:

18.04, 18.71, 18.92, 19.25, 19.29, 19.44, 19.77, 20.17, 20.33, 20.50, 20.72, 21.12, 21.41, 21.77, 21.81, 22.11, 22.43, 22.85, 23.00, 23.71

Since there are 20 values, the median is the average of the 10th and 11th values:

Median = (20.50 + 20.72) / 2

Median ≈ 20.61 (rounded to two decimal places)

b. 10% Trimmed Mean:

To compute the 10% trimmed mean, you first remove the lowest and highest 10% of the data and then calculate the mean of the remaining values:

10% of 20 values = 0.10 * 20 = 2 values

Remove the two lowest values (18.04 and 18.71) and the two highest values (23.00 and 23.71):

Trimmed Mean = (18.92 + 19.25 + ... + 22.85) / 16

Trimmed Mean ≈ 20.70 (rounded to two decimal places)

c. Dot Plot of the Absorbency Data:

I'm unable to create a visual plot in this text-based format, but you can create a dot plot by plotting each data point as a dot along a number line. The dots should be stacked vertically when there are multiple data points with the same value.

d. Outliers:

To determine if there are outliers, we can compare the mean, median, and trimmed mean. If there are outliers, they tend to affect the mean more than the median and trimmed mean. In this case, the mean is approximately 20.98, the median is approximately 20.61, and the trimmed mean is approximately 20.70.

Since the mean is slightly higher than the median and trimmed mean, there may be some positive outliers in the data, but the difference is not substantial. Further analysis, such as a box plot or identifying specific data points that deviate significantly from the central tendency, can provide a more definitive assessment of outliers.

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Rewritten by : Barada

The Sample Mean of the absorbency values is 20.7835 and the Sample Median of the absorbency values is 20.61.

How to calculate the Sample Mean and Median for Fiber Absorbency Data:

Sample Mean:

Sum of absorbency values:

= 18.71 + 21.41 + 20.72 + 21.81 + 19.29 + 22.43 + 20.17 + 23.71 + 19.44 + 20.50 + 18.92 + 20.33 + 23.00 + 22.85 + 19.25 + 21.77 + 22.11 + 19.77 + 18.04 + 21.12

Sum = 415.67

Sample Mean = Sum / Number of observations

= 415.67 / 20

= 20.7835

Sample Median Calculation:

Arrange values in ascending order:

18.04, 18.71, 18.92, 19.25, 19.29, 19.44, 19.77, 20.17, 20.33, 20.50, 20.72, 21.12, 21.41, 21.77, 21.81, 22.11, 22.43, 22.85, 23.00, 23.71

Since there are 20 observations, the median is the average of the 10th and 11th values.

Median = (20.50 + 20.72) / 2

= 20.61.

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