Question

A small liberal arts college in the Northeast has 350 freshmen. One hundred twenty of the freshmen are education majors. Suppose seventy freshmen are randomly selected (without replacement).

Step 2 of 2: Find the standard deviation of the number of education majors in the sample. Round your answer to two decimal places, if necessary.

Answer 1

The standard deviation of the number of education majors in a sample of 70 freshmen from a college with a freshman class of 350, where 120 are education majors, is approximately 3.01.

Step-by-step explanation:

To find the standard deviation of the number of education majors in a sample of 70 freshmen from a college with 350 freshmen, of which 120 are education majors, we can use the formula for the standard deviation of a hypergeometric distribution because the sample is being drawn without replacement.

The formula for the standard deviation (σ) of a hypergeometric distribution is:

σ = √[n * (K/N) * (1 - K/N) * (N-n)/(N-1)]

Where:

• n is the sample size (70 freshmen)
• K is the number of successes in the population (120 education majors)
• N is the population size (350 freshmen)

Plugging in the numbers gives:

σ = √[70 * (120/350) * (1 - 120/350) * (350-70)/(350-1)]

Calculating further, we get:

σ = √[70 * 0.34286 * 0.65714 * 280/349]

σ = √[16.709 * 0.65714 * 0.80229]

σ = √[9.067]

σ ≈ 3.01 (after rounding to two decimal places)

Therefore, the standard deviation of the number of education majors in the sample is approximately 3.01.