At the end of every school year, the school administers a reading test to a sample of 36 third graders.

At the end of every school year, the school
administers a reading test to a sample of 36 third graders. The school system
has 20,00 third graders, half boys and half girls. The
results from last year's test are shown in the table below.

Stratum	Mean score	Standard deviation
Boys	60	10.27
Girls	55	6.66

 
This year, the researchers plan to use a stratified
sample, with one stratum consisting of boys and the other, girls. Use the
results from last year to answer the following questions? To maximize
precision, how many sampled students should be boys and how many should be
girls?

Solution: 

To maximize precision in the stratified sample, you should allocate the sample size based on the variances of the two strata. In this case, the strata are boys and girls. Here's how you can calculate the sample sizes for boys and girls:

Calculate the variance of each stratum:

Variance of boys = Standard deviation of boys squared = 10.27^2

Variance of girls = Standard deviation of girls squared = 6.66^2

Calculate the proportion of boys and girls in the population:

Proportion of boys = Proportion of girls = 1/2 (since half are boys and half are girls)

Calculate the allocation fraction for each stratum:

Allocation fraction for boys = Variance of boys / (Variance of boys + Variance of girls)

Allocation fraction for girls = Variance of girls / (Variance of boys + Variance of girls)

Calculate the sample sizes for boys and girls:

Sample size for boys = Allocation fraction for boys * Total sample size

Sample size for girls = Allocation fraction for girls * Total sample size

Using the provided information:

Variance of boys = 10.27^2 ≈ 105.4249

Variance of girls = 6.66^2 ≈ 44.3556

Proportion of boys = Proportion of girls = 1/2

Allocation fraction for boys = 105.4249 / (105.4249 + 44.3556) ≈ 0.7037

Allocation fraction for girls = 44.3556 / (105.4249 + 44.3556) ≈ 0.2963

Total sample size = 36

Sample size for boys = 0.7037 * 36 ≈ 25.3332

Sample size for girls = 0.2963 * 36 ≈ 10.6668

To maximize precision, you should round the sample sizes to the nearest whole numbers:

Sample size for boys: 25

Sample size for girls: 11

Therefore, to maximize precision, you should sample 25 boys and 11 girls.