Question

An engineer is working on designing storage space for two preschool classrooms with shelves for students to store their things, and he is trying to determine how high the top shelf can be so that at least a good number of them will be able to reach it. He consults the data that he has for preschoolers in this particular population, and finds that the average preschooler is 45 inches tall with a standard deviation of 4 inches. (a) What percentage of preschoolers in the population are more than 47 inches tall?
(b) The target class size for each classroom is 20 students, so 40 students would be using the shelves. Find the sampling distribution for the proportion of students out of 40 (randomly assigned preschoolers that are more than 47 inches tall? (c) You applied the Central Limit Theorem above. Was your sample size large enough to do so? Justify your answer. (d) If the engineer designs the classroom so that the top shelf is reachable by children at least 47 inches tall, what is the likelihood that fewer than 30% of the 40 children will be able to reach the top shelf?

Answer 1

Mean, = 45 inches

Standard deviation, = 4 inches

(a) Percentage of preschoolers in the population more than 47 inches tall = P(X > 47)*100%

= P{Z > (47 - 45)/4}*100%

= 30.85%

(b) The sampling distribution for the proportion of students out of 40 that are more than 47 inches tall is normally distributed with mean 45 inches and standard error = 4/√40 = 0.6325 inches

(c) Yes the sample size was large enough. A sample size greater than or equal to 30 is enough to use Central Limit Theorem

(d) Fewer than 30% of 40 -> atmost 11

This can be solved using Binomial distribution where n = 40, p = 0.3085

To find P(X ≤ 11) = 0.3949