Statistics
101Centrality,
Spreads, Normality, and Chebyshev's RuleSolution
to Problem 3
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101Centrality,
Spreads, Normality, and Chebyshev's RuleSolution
to Problem 3
Problem 3.
Below is a table of weekly referrals from a forum to a website. The referral pattern is quite typical for many forums.
Weekly Referrals from a Forum to a Website for the first 12 weeks of 2006 Week
1/01/2006-1/07/2006
1/08/2006-1/14/2006
1/15/2006-1/21/2006
1/22/2006-1/28/2006
1/29/2006-2/04/2006
2/05/2006-2/11/2006
2/12/2006-2/18/2006
2/19/2006-2/25/2006
2/26/2006-3/04/2006
3/05/2006-3/11/2006
3/12/2006-3/18/2006
3/19/2006-3/25/2006Referrals
41
6
9
0
0
0
0
4
5
1
1
1For the above population data:
(a) Compute the mean,
(b) Determine the median,
(c) Compute the variance,
(d) Compute the standard deviation,
(e) Compute the corresponding z-score for each weekly referral total,
(f) Verify that for the z-score intervals (-1.5,1.5), (-2,2), and (-3,3) the data set conforms to Chebyshev's Rule,
(g) Construct a Stem-and-Leaf Display for the referral data, and
(h) Construct, using Minitab, a frequency histogram with:(i) intervals 3-referrals wide, over the domain 0-to-42,
(ii) first value in Minitab's lower interval-definition box to be left endpoint of first interval,
(iii) mean and median marked by vertical lines, and
(iv) z-scores below each x-axis value and the mean.
Solution to Problem 3: Centrality, Spreads, Normality, and Chebyshev's Rule
To compute the mean and the variance, it is convenient to compute first these two values:
41 + 6 + 9 + 0 + 0 + 0 + 0 + 4 + 5 + 1 + 1 + 1 68 412 + 62 + 92 + 02 + 02 + 02 + 02 + 42 + 52 + 12 + 12 + 12 1,842 (a) We then compute the mean.
(68)/12 5.67 (b) Then determine the median. As there are an even number of data, we take the average of the middle-two values. We have:
median
(1+1)/2
(c) Then compute the variance.
(1,842 -(68)2/12)/12 121.39 (d) Then compute the standard deviation.
(121.39)1/2 11.02 (e) Then compute the corresponding z-score for each weekly visitation.
Data Formula for z-scores z-scores xi zi 41
6
9
0
0
0
0
4
5
1
1
1(41-5.67)/11.02
(6-5.67)/11.02
(9-5.67)/11.02
(0-5.67)/11.02
(0-5.67)/11.02
(0-5.67)/11.02
(0-5.67)/11.02
(4-5.67)/11.02
(5-5.67)/11.02
(1-5.67)/11.02
(1-5.67)/11.02
(1-5.67)/11.023.21
0.03
0.30
-0.51
-0.51
-0.51
-0.51
-0.15
-0.06
-0.42
-0.42
-0.42
(f) Then verify that the percentages of the population with z-scores within the intervals (-1.5,1.5), (-2,2), and (-3,3) conform to Chebyshev's Rule. We have:
z-score Interval Number of z-scores zi within Interval, from (e) above Computation of Percentage of z-scores within Interval Percentage of Population with z-scores within Interval Chebyshev's Formula: (1 - 1/z2)
(100%)
Chebyshev's Percentage (-1.5,1.5) 11 (11/12) 91.7% (1 - 1/1.52) 55.6% (-2,2) 11 (11/12) 91.7% (1 - 1/22) 75% (-3,3) 11 (11/12) 91.7% (1 - 1/32) 88.9% and from this table, we have:
The percentage of the population with z-scores within the interval (-1.5,1.5)
55.6%
The percentage of the population with z-scores within the interval (-2,2)
The percentage of the population with z-scores within the interval (-3,3)
(g) Then construct a Stem-and-Leaf Display for the visitation data. We use Minitab. We drag across the first column of our data table at the top of the page and paste the contents into the top 13 cells (the title row and rows 1-12) of the first column in Minitab's worksheet. The data table's second column goes into the top 13 cells of Minitab's second column. (For more detailed instructions, see Problem 1: Displaying Quantitative Data.) With the data transferred, we construct the stem-and-leaf display. (For instructions, see Stem-and-Leaf Construction Using Minitab Release 13.) We have:
(h) Construct, using Minitab, a frequency histogram with:
(i) intervals 3-referrals wide, over the domain 0-to-42,
(ii) first value in Minitab's lower interval-definition box to be left endpoint of first interval,
(iii) mean and median marked by vertical lines, and
(iv) z-scores below each x-axis value and the mean.We follow the basic instructions of Histogram Construction Using Minitab Release 13. Requirements (i) and (ii) impose the following selections in the Type of Intervals and Definition of Intervals sections of the Histogram Options dialog box.
After clicking the OK button in the Histogram Options dialog box and and again in the Histogram dialog box, we use the Tool bar to add the mean and median lines, the z-scores, and labels. We have:
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