A computer scanner can read a bar code on a package correctly 97% of the time. One company monitors the accuracy of the scanner by randomly sampling packages and verifying that each package has been correctly scanned. Random samples of size n = 25, 50, 100, and 200 have recently been taken with the following results.
Package Sampling Table
Sample Size, n Number Correctly Scanned
25 24
50 49
100 95
200 193
To complete this activity, use Excel to:
Calculate the sample proportion for each sample size.
Calculate the single-proportion sampling error for each sample size.
Calculate the probability of finding 198 correctly scanned packages, for a sample of size n=200.
To calculate the sample proportion for each sample size, we divide the number of packages correctly scanned by the sample size.
Sample Size, n Number Correctly Scanned Sample Proportion
25 24 24/25 = 0.96
50 49 49/50 = 0.98
100 95 95/100 = 0.95
200 193 193/200 = 0.965
To calculate the single-proportion sampling error for each sample size, we subtract the sample proportion from the population proportion (97%) and divide it by the square root of the product of the sample proportion and the complementary proportion.
Sample Size, n Number Correctly Scanned Sample Proportion Sampling Error
25 24 0.96 (0.97 - 0.96) / sqrt(0.97 * 0.03) = 0.033
50 49 0.98 (0.97 - 0.98) / sqrt(0.97 * 0.03) = -0.033
100 95 0.95 (0.97 - 0.95) / sqrt(0.97 * 0.03) = 0.067
200 193 0.965 (0.97 - 0.965) / sqrt(0.97 * 0.03) = 0.034
To calculate the probability of finding 198 correctly scanned packages for a sample of size n=200, we can use the binomial distribution formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
n is the sample size (200)
k is the number of correctly scanned packages (198)
p is the population proportion (97% or 0.97)
P(X=198) = (200 choose 198) * 0.97^198 * (1-0.97)^(200-198)
Using Excel or a calculator, we can calculate this probability:
P(X=198) = (200! / (198! * (200-198)!)) * 0.97^198 * (1-0.97)^(200-198)
P(X=198) ≈ 0.0039
Therefore, the probability of finding exactly 198 correctly scanned packages for a sample of size n=200 is approximately 0.0039, or about 0.39%.