Question

A survey of students at a large university found that 82% had purchased textbooks from an off-campus vendor at least once during their college career. If 45 students are randomly sampled, what is the probability that at least 40 students have purchased textbooks from an off-campus vendor at least once during their college career? g

Answer 1

Probability that at least 40 students have purchased textbooks from an off-campus vendor at least once during their college career is 0.0688.

Explanation:

We are given that a survey of students at a large university found that 82% had purchased textbooks from an off-campus vendor at least once during their college career.

Also, 45 students are randomly sampled.

Let
`\hat p` = sample proportion of students who have purchased textbooks from an off-campus vendor at least once during their college career.

The z-score probability distribution for sample proportion is given by;

Z =
`\frac{\hat p- p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample proportion =
`(40)/(45)` = 0.89

p = population proportion of students who had purchased textbooks from an off-campus vendor at least once during their college career = 82%

n = sample of students = 45

Now, probability that at least 40 students have purchased textbooks from an off-campus vendor at least once during their college career is given by = P(
`\hat p`
`\geq` 0.89)

P(
`\hat p`
`\geq` 0.89) = P(
`\frac{\hat p- p}{\sqrt{(\hat p(1-\hat p))/(n) } }`
`\geq`
`\frac{0.89-0.82}{\sqrt{(0.89(1-0.89))/(45) } }` ) = P(Z
`\geq` 1.50) = 1 - P(Z < 1.50)

= 1 - 0.9332 = 0.0688

The above probability is calculate by looking at the value of x = 1.50 in the z table which ha an area of 0.9332.

Therefore, the required probability is 0.0688.