The director of admissions at Kinzua University in Nova Scotia estimated the distribution of student admissions for the fall semester on the basis of past experience.
Admissions Probability
1,080 0.5
1,340 0.2
1,660 0.3
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(a) What is the expected number of admissions for the fall semester? (Round your answers to the nearest whole number.)
Expected number of admissions
(b) Compute the variance and the standard deviation of the number of admissions. (Round your answers to 2 decimal places.)
Variance
Standard deviation
To calculate the expected number of admissions for the fall semester, you need to multiply each number of admissions by their respective probabilities and then sum up the results.
(a) Expected number of admissions:
Expected number of admissions = (1,080 * 0.5) + (1,340 * 0.2) + (1,660 * 0.3)
Expected number of admissions = 540 + 268 + 498
Expected number of admissions = 1306
So, the expected number of admissions for the fall semester is 1306.
To calculate the variance and standard deviation of the number of admissions, you need to calculate the squared difference between each number of admissions and the expected number of admissions. Then, multiply each squared difference by their respective probabilities and sum up the results. The variance is the sum of these squared differences, and the standard deviation is the square root of the variance.
(b) Variance:
Variance = [(1,080 - 1306)^2 * 0.5] + [(1,340 - 1306)^2 * 0.2] + [(1,660 - 1306)^2 * 0.3]
Variance = [(-226)^2 * 0.5] + [(-34)^2 * 0.2] + [354^2 * 0.3]
Variance = (51176 * 0.5) + (1156 * 0.2) + (125316 * 0.3)
Variance = 25588 + 231.2 + 37594.8
Variance ≈ 63414
Therefore, the variance of the number of admissions is approximately 63414.
Standard deviation:
Standard deviation = √Variance
Standard deviation = √63414
Standard deviation ≈ 251.87
So, the standard deviation of the number of admissions is approximately 251.87.