Speaking of Statistics(12) - Assignment 1 – Suggested Solution
1. Below we list several variables. Which of these variables are quantitative and which are
qualitative? Explain.
a. The dollar amount on an accounts receivable invoice. ? Quantitative b. The net profit for a company in 2005. ? Quantitative c. The stock exchange on which a company’s stock is traded. ? Qualitative d. The national debt of the United States in 2005. ? Quantitative e. The advertising medium (radio, television, or print) used to promote a product. ? Qualitative
2. Classify each of the following qualitative variables as ordinal or nominative. Explain your
answers.
Qualitative Variables The Corresponding Categories
Statistics course letter grade A B C D F ?Ordinal Personal computer ownership Yes No ? Nominal Restaurant rating ***** **** *** ** * ? Ordinal Incoming tax filing status Married filing jointly; Married filing separately; Single; Head
of household; Qualifying widow(er) ? Nominal
3. Forbes magazine publishes the Forbes Platinum 400 –a list of the ―Best Big Companies in
America‖ as selected by the magazine’s writers and editors. Table above gives the best companies in the retailing industry as given in this list on the Forbes website on March 16, 2005. (Data in the last page).
b. Organize data of the profit margin percentages into a frequency distribution.
Number of class K=6 (Since 2?n where n is the total number of data = 35.) The class length is calculated as Class 1.1-4.9 5.0-8.8 8.9-12.7 12.8-16.6 16.7-20.5 20.6-24.4 Frequency 14 17 3 0 0 1 Relative Frequency 0.40 0.49 0.09 0.00 0.00 0.03 Percentage Frequency 40.0% 48.6% 8.6% 0.0% 0.0% 2.9% Krange24.0?1.1??3.81?3.9 K6
a. Construct a stem-and leaf display of the return on capital percentages for the retailers in Table.
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Find the median and then describe the distribution of the return.
The median is the 18th measurement, 15.4.
The data are widely spread out, with majority falling in the range [8.3, 22.8]. The distribution is somewhat symmetric, with extreme values at both ends.
c. Construct a histogram of the profit margin percentages.
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4. In order to control costs, a company wishes to study the amount of money its sales force
spends entertaining clients. The following is a random sample of six entertaining expenses (dinner costs for four people) from expense reports submitted by members of the sales force $157 $132 $109 $145 $125 $139
a. Show the equivalence of the following two formulas:
Solutions:
,
.
s2???????
2(x?x)?in?11(xi?x)2?n?11[(x1?x)2?(x2?x)2?...?(xn?x)2]n?12221222[(x1?2x1x?x)?(x2?2x2x?x)?...?(xn?2xnx?x)]n?1221222[(x1?x2?...?xn)?(2x1x?2x2x?...?2xnx)?(x?...?x)]n?12????xx1???ii2??(?xi?2(?xi)?n??n?1?nn??????1?12?2??x? x?in?i?n ?1???
b. Calculate x, s, and s for the expense data, use the two different formulas in (a) to calculates.
Solutions x?222157?132?109?145?125?139?134.5
6(157 134.5)2?(132 134.5)2???(139 134.5)2s??276.7 , s?276.7?16.63
6 1
Or using another formula:
?x2i?1572?1322?1092?1452?1252?1392?109,925
2??x?i?(157?132?109?145?125?139)2?(807)2?651,249
s2?
1?11?12??2?109,925–(651,249)?276.7??x? x??ii????6?n ?1?n?6–1?
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5. On February 15, 2005, the Gallup Organization released the results of a Gallup UK poll regarding Internet usage in Great Britain. Each of 1,009 randomly selected UK adults was asked to respond to the following question:
How much time, if at all, do you personally spend using the Internet-more than an hour a day, up to one hour a day, a few times a week, a few times a month or less, or never?
The poll’s results were as follows: More than an hour a day (22%); Up to an hour a day (14%); A few times a week (15%); A few times a month or less (10%); Never (39%). Use these data to construct a bar chart and a pie chart.
Bar Chart:
Pie Chart:
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6. Thirteen internists in the Midwest are randomly selected, and each internist is asked to report last year’s income. The incomes obtained (in thousands of dollars) are
127 132 138 141 144 146 152 154 162 171 177 192 241 Find:
There are 13 measurements in total (n=13). They are already in ascending order. a. The 90th percentile.
Calculate the index of the 90th
percentile: i?90100?13?11.7 Round 11.7 up to 12, hence the 90th percentile is the 90th measurement 192.
b. The median.
The median is equivalent to the 50th
percentile: i?50100?13?6.5 Round 6.5 up to 7, hence the median is the 7th measurement 152.
c. The first quartile.
We know the index of the first quartile can be found as: i?25100?13?3.25 Round 3.25 up to 4, hence the first quartile is the 4th measurement 141.
d. The third quartile
We know the index of the third quartile can be found as: i?75100?13?9.75Round 9.75 up to 10, hence the third quartile is the 10th measurement 171.
e. The 10th percentile.
Calculate the index of the 10th
percentile: i?10100?13?1.3 Round 1.3 up to 2, hence the 10th percentile is the 2nd measurement 132.
f. The interquartile range.
Interquartile range is the difference between first and third quartile
IQR?Q3?Q1?171?141?30
g. Develop a five-number summary and a box-and-whiskers display.
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