Question

28 The following is a sample of 25 measurements. L02098 7 6 6 11 8 9 11 9 10 8 7 7 5 9 10 7 7 7 7 9 12 10 10 8 6 a. Compute , s2, and s for this sample b. Count the number of measurements in the intervals e ± s, x ± 2s, and x ± 3s. Express each count as a percentage of the total number of measurements. c. Compare the percentages found in part b with the percentages given by the empirical rule and Chebyshev's rule. d. Calculate the range and use it to obtain a rough approx- imation for s. Does the result compare favorably with the actual value for s found in part a?

Answer 1

(a)

Following table shows the calculations:

Sample size: n=25

Now,

Mean:

The variance:

Standard deviation:

(b)

Following is the ordered data set:

10 10 10 10 12

First interval:

Out of 25 data values, 18 lies in the above interval so the required percentage is

Second interval:

Out of 25 data values, 24 lies in the above interval so the required percentage is

Third interval:

Out of 25 data values, 25 lies in the above interval so the required percentage is

(c)

According to empirical rule 68% data values lies within one standard deviation of mean, 95% data values lies within two standard deviations of mean and 99.7% data values lies within three standard deviations of mean.

All the percentages of part b are greater than percentages given by empirical rule.

------------

Chebyshev's rule:

According to Chebyshev's rule at least 0% data values lies within one standard deviation of mean, at least 75% data values lies within two standard deviations of mean and at least 88.9% data values lies within three standard deviations of mean.

All the percentages of part b are as specified by part c.

(d)

The range is

Range = Maximum - Minimum = 12 - 5 = 7

According to range rule the standard deviation is approximately equal to one fourth of the range of the data so

The standard deviation found in part (a) is greater than the above estimated standard deviation.