Statistics
101Centrality,
Spreads, Normality, and Chebyshev's RuleSolution
to Problem
1
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101Centrality,
Spreads, Normality, and Chebyshev's RuleSolution
to Problem
1
Problem 1.
Below is a table of weekly visitation to a web page.
Weekly Visitation of a Web Page 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
3/19/2006-3/25/2006Visitation
386
455
418
360
321
383
363
342
444
395
436
407For 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 visitation,
(f) Verify that the percentages of the population with z-scores within the intervals (-1.5,1.5) and (-2,2) conform to Chebyshev's Rule,
(g) Construct a Stem-and-Leaf Display for the visitation data, and
(h) Construct, using Minitab, a frequency histogram with:(i) intervals 20-visitations wide, over the domain 320-to-460,
(ii) first value in Minitab's lower interval-definition box to be left endpoint of first interval (here, tick marks and values at the endpoints of each interval),
(iii) mean and median marked by vertical lines, and
(iv) z-scores below each x-axis value and the mean.Solution to Problem 1: Centrality, Spreads, Normality, and Chebyshev's Rule
To compute the mean and the variance, it is convenient to compute first these two values:
386 + 455 + 418 + 360 + 321 + 383 + 363 + 342 + 444 + 395 + 436 + 407 4,710 3862 + 4552 + 4182 + 3602 + 3212 + 3832 + 3632 + 3422 + 4442 + 3952 + 4362 + 4072 1,867,714 (a) We then compute the mean.
(4,710)/12 392.5 (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
(386+395)/2
(c) Then compute the variance.
(1,867,714 -(4,710)2/12)/12 1,586.58 (d) Then compute the standard deviation.
(1,586.58)1/2 39.83 (e) Then compute the corresponding z-score for each weekly visitation.
Data Formula for z-scores z-scores xi zi 386
455
418
360
321
383
363
342
444
395
436
407(386-392.5)/39.83
(455-392.5)/39.83
(418-392.5)/39.83
(360-392.5)/39.83
(321-392.5)/39.83
(383-392.5)/39.83
(363-392.5)/39.83
(342-392.5)/39.83
(444-392.5)/39.83
(395-392.5)/39.83
(436-392.5)/39.83
(407-392.5)/39.83-0.16
1.57
0.64
-0.82
-1.80
-0.24
-0.74
-1.27
1.29
0.06
1.09
0.36
(f) Then verify that the percentages of the population with z-scores within the intervals (-1.5,1.5) and (-2,2) 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) 7 (7/12) 58.3% (1 - 1/1.52) 55.6% (-2,2) 12 (12/12) 100% (1 - 1/22) 75% 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)
(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) Then construct, using Minitab, a frequency histogram with:
(i) intervals 20-visitations wide, over the domain 320-to-460,
(ii) first value in Minitab's lower interval-definition box to be left endpoint of first interval (here, tick marks and values at the endpoints of each 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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