Answer :
Answer:
Probability that at least 40 students have purchased textbooks from an off-campus vendor at least once during their college career is 0.0688.
Step-by-step 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 [tex]\hat p[/tex] = 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 = [tex]\frac{\hat p- p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ~ N(0,1)
where, [tex]\hat p[/tex] = sample proportion = [tex]\frac{40}{45}[/tex] = 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( [tex]\hat p[/tex] [tex]\geq[/tex] 0.89)
P( [tex]\hat p[/tex] [tex]\geq[/tex] 0.89) = P( [tex]\frac{\hat p- p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] [tex]\geq[/tex] [tex]\frac{0.89-0.82}{\sqrt{\frac{0.89(1-0.89)}{45} } }[/tex] ) = P(Z [tex]\geq[/tex] 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.