If people choose lottery numbers at random, the last digit should be equally likely to be any of the digits from 0 to 9. Let p measure the proportion of choices that end with the digit 7. If choices are random, we would expect p = 0.10, but if people have a special preference for numbers ending in 7 the proportion will be greater than 0.10. Suppose that we test this by asking a random sample of 20 people to give a three-digit lottery number and find that four of the numbers have 7 as the last digit. Figure 5.22 shows a randomization distribution of proportions for 5000 simulated samples under the null hypothesis H₀: p = 0.10.

Figure 5.22

(a) Use the sample proportion p̂ = 0.20 and a standard error estimated from the randomization distribution to compute a standardized test statistic.

(b) Use the normal distribution to find a p-value for an upper tail alternative based on the test statistic found in part (a).

(c) Compare the p-value obtained from the normal distribution in part (b) to the p-value shown for the randomization distribution. Explain why there might be a discrepancy between these two values.

Transcribed Image Text:

1500 Left Tail D Two-Tail D Right Tail Number of Randomization Samples = 5000 Mean = 0.101 Std. Dev. = 0.067 1250 1000 750 0.137 500 250 0.00 0.05 0.10 0.25 0.15 0.20 0.30 0.35 0.40 0.45 null = 0.100 0.200