Imprints Galore buys T-shirts (to be imprinted with an item of the customer’s choice) from a manufacturer who guarantees that the shirts have been inspected and that no more than 1% are imperfect in any way. The shirts arrive in boxes of 12. Let x be the number of imperfect shirts found in any one box.

a. List the probability distribution and draw the histogram of x.

b. What is the probability that any one box has no imperfect shirts?

c. What is the probability that any one box has no more than one imperfect shirt?

d. Find the mean and standard deviation of x.

e. What proportion of the distribution is between

   and   ?

f. What proportion of the distribution is between

  2 and   2?

g. How does this information relate to the empirical rule and Chebyshev’s theorem? Explain.

h. Use a computer to simulate Imprints Galore’s buying 200 boxes of shirts and observing x, the number of imperfect shirts per box of 12. Describe how the information from the simulation compares to what was expected (answers to parts a–g describe the expected results).

i. Repeat part h several times. Describe how these results compare with those of parts a–g and with part h.

MINITAB (Release 14)
a.
Choose: Calc  Make Patterned Data  Simple Set of Numbers . . .
Enter: Store patterned data in: C1 From first value: 1 (see note)
To last value: 12 In steps of: 1  OK

c. Continue with the binomial probability MINITAB commands on page 292, using n  12, p  0.01, and C2 for optional storage.
Choose: Graph  Scatterplot  Simple  OK Enter: Y variables: C2 X variables: C1 Select: Data view: Data Display: Area  OK The graph is not a histogram, but can be converted to a histogram by double clicking on ‘area’ of graph.
Select: Options Select: Step  OK  OK h. Continue with the cumulative binomial probability MINITAB commands on page 292, using n  12, p  0.01, and C3 for optional storage.
Choose: Calc  Random Data  Binomial Enter: Generate: 200 rows of data Store in column C4 Number of trials: 12 Probability: .01  OK Choose: Stat  Tables  Cross Tabulation Enter: Categorical variables: For rows: C4 Select: Display: Total percents OK Choose: Calc  Column Statistics Select: Statistic: Mean Enter: Input variable: C4  OK Choose: Calc  Column Statistics Select: Statistic: Standard deviation Enter: Input variable: C4  OK Continue with the histogram MINITAB commands on page 61, using the data in C4 and selecting the options: percent and midpoint with intervals 0:12/1.
Note: The binomial variable x cannot take on the value 1. The use of 1 (the next would-be class midpoint to left of 0) allows MINITAB to draw the histogram of a probability distribution. Without 1, PLOT will draw only half of the bar representing x  0.

Excel a.
Enter: 0,1,2, . . . ,12 into column A Continue with the binomial probability Excel commands on pages 292–293, using n  12 and p  0.01.
Activate columns A and B; then continue with:
Choose: Chart Wizard  Column  1st picture(usually) 
Next  Series Choose: Series 1  Remove Enter: Category (x)axis labels: (A1:A13 or select ‘x value’ cells)
Choose: Next  Finish Click on:Anywhere clear on the chart —use handles to size so x values fall under corresponding bars Continue with the cumulative binomial probability Excel commands on pages 292–293, using n  12, p  0.01, and column C for the activated cell.
h.
Choose: Tools  Data analysis  Random Number Generation  OK Enter: Number of Variables: 1 Number of Random Numbers: 200 Distribution: Binomial p Value  0.01 Number of Trials  12 Select: Output Options: Output Range Enter (D1 or select cell)  OK Activate the E1 cell, then:
Choose: Insert function, fx  Statistical  AVERAGE  OK Enter: Number 1: D1:D200  OK Activate the E2 cell, then:
Choose: Insert function, fx  Statistical  STDEV  OK Enter: Number 1: D1:D200  OK Continue with the histogram Excel commands on pages 61–62, using the data in column D and the bin range in column A.
TI-83/84 Plus a.
Choose: STAT  EDIT  1:Edit Enter: L1: 0,1,2,3,4,5,6,7,8,9,10,11,12 Choose: 2nd QUIT  2nd DISTR  0:binompdf(
Enter: 12, 0.01)  ENTER Choose: STO0  L2  ENTER Choose: 2nd  STAT PLOT  1:Plot1 Screen capture 5.5A Choose: WINDOW Enter: 0, 13, 1, .1, .9, .1, 1 Choose: TRACE   
c.
Choose: 2nd  DISTR  A:binomcdf(
Enter: 12, 0.01)
Choose: STO0  L3  ENTER STAT  EDIT  1:Edit h.
Choose: MATH  PRB  7:randBin(
Enter: 12, .01, 200) (takes a while to process)
Choose: STO0  L4  ENTER Choose: 2nd LIST  Math  3:mean(
Enter: L4 Choose: 2nd LIST  Math  7:StdDev(
Enter: L4 Continue with the histogram TI-83/84 commands on pages 62–63, using the data in column L4 and adjusting the window after the initial look using ZoomStat.