Police are tested for their ability to correctly recognize and identify a suspect based on a witness or victim’s verbal description of the suspect. Scores on the identifi­cation test range from 0 to 100 (perfect score).

Three cities in Massachusetts are under study. The mean score for all police in the three cities is requested. However, funding will only permit m = 150 police to be tested. There is no preliminary study to estimate sample standard deviation of scores in each city. City A has N₁ = 183 police, City B has N₂ = 371 police, and City C has N₃ = 255 police.

If we have no preliminary study for sample standard deviations, we can use a method called proportional sampling (also called representative sampling). The sample size n1 for each stratum (city police department) is given by

In most cases n_i is not a whole number so we round to the nearest whole number.

Remember our total sample size is m = n₁ + n₂ + n₃.

(a) Use the method of proportional sampling to compute n₁, n₂, and n₃, the size of the random sample to be taken from each of the three cities. Round each n_i to the nearest whole number and make sure m = n₁ + n₂ + n₃.

(b) Suppose you actually conducted the speci­fied number of tests in each city and obtained the following result: x̅₁ = 96 is the mean test score from city A, x̅₂ = 85 is the mean test score from City B, and x̅₃ = 88 is the mean test score from City C. Use the weighted average

To get your best estimate for the population mean test score of all police in all three cities.

R. N+N2+N