Question 7.81

image 51. You can use a table of random digits to simulate sampling from a population. Suppose that 60% of the population bought a lottery ticket in the last 12 months. We will simulate the behavior of random samples of size 40 from this population.

  1. Let each digit in the table stand for one person in this population. Digits 0 to 5 stand for people who bought a lottery ticket, and 6 to 9 stand for people who did not. Why does looking at one digit from Table 7.1 (page 298) simulate drawing one person at random from a population with 60% “yes”?
  2. Each row in Table 7.1 contains 40 digits. So the first 10 rows represent the results of 10 samples. How many digits between 0 and 5 does the top row contain? What is the percentage of “yes” responses in this sample? How many of your 10 samples overestimated the population proportion of 60%? How many underestimated it? (You could program a computer to continue this process, say, 1000 times, to produce a pattern like that in Figure 7.6 on page 316.)

51.

(a) Each digit in the table has 1 chance in 10 to be any of the 10 possible digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. So in the long run, 60% of the digits we encounter will be 0, 1, 2, 3, 4, or 5, and 40% will be 6, 7, 8, or 9.

(b) Line 101 contains 29 digits 0 to 5. This stands for a sample with “yes” responses. If we use lines 101 to 110 to simulate 10 samples, the counts of “yes” responses are 29, 24, 23, 23, 20, 24, 23, 19, 24, and 18. Thus, three samples are exactly correct ( ), one overestimates, and six underestimate.