Statistics and Probability Problems With Solutions

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Problem 1

For boys, the average number of absences in the first grade is 15 with a standard deviation of 7; for girls, the average number of absences is 10 with a standard deviation of 6.

In a nationwide survey, suppose 100 boys and 50 girls are sampled. What is the probability that the male sample will have at most three more days of absences than the female sample?

(A) 0.025
(B) 0.035
(C) 0.045
(D) 0.055
(E) None of the above

Solution

The correct answer is B. The solution involves four steps.

Find the mean difference (male absences minus female absences) in the population.
μ_d = μ₁ - μ₂ = 15 - 10 = 5
Find the standard deviation of the difference.
σ_d = sqrt( σ₁² / n₁ + σ₂² / n₂ )
σ_d = sqrt(7²/100 + 6²/50) = sqrt(49/100 + 36/50) = sqrt(0.49 + .72) = sqrt(1.21) = 1.1
Find the z-score that produced when boys have three more days of absences than girls. When boys have three more days of absences, the number of male absences minus female absences is three. And the associated z-score is
z = (x - μ)/σ = (3 - 5)/1.1 = -2/1.1 = -1.818
Find the probability. This problem requires us to find the probability that the average number of absences in the boy sample minus the average number of absences in the girl sample is less than 3. To find this probability, we enter the z-score (-1.818) into Stat Trek's Normal Distribution Calculator. We find that the probability of a z-score being -1.818 or less is about 0.035.