Question

At a certain university, 50% of all entering freshmen planned to major in a STEM (science, technology, engineering, mathematics) discipline. A sample of 36 freshmen is selected. What is the probability that the proportion of freshmen in the sample is between 0.482 and 0.580? Write the answer as a number to the 4th decimal (0.1234).

The intended steps are as follows:

Step 1: Check to see that the conditions np ≥ 10 and n(1− p) ≥ 10 are
both met. If so, it is appropriate to use the normal curve.
Step 2: Find the mean U_p and standard deviation a_p.
Step 3: Sketch a normal curve and shade in the area to be found.
Step 4: Find the area using the TI-84 PLUS.

Answer 1

Solution:

Given: At a certain university, 50% of all entering freshmen planned to major in a STEM (science, technology, engineering, mathematics) discipline.

thus p = 0.5

Sample size = n = 36

We have to find  the probability that the proportion of freshmen in the sample is between 0.482 and 0.580.

That is:

Step 1) Check to see that the conditions np ≥ 10 and n(1− p) ≥ 10 are both met. If so, it is appropriate to use the normal curve.

n*p = 36 * 0.5 = 18 > 10

and

n*(1-p)= 36 *(1-0.5) = 36 * 0.5 = 18 > 10

Both conditions are met, thus  it is appropriate to use the normal curve.

Step 2) Find the mean   and standard deviation .

Step 3: Sketch a normal curve and shade in the area to be found.

Step 4: Find the area using the TI-84 PLUS.

Use following steps in TI84 PLUS calculator.

Press 2ND

Press VARS
select normalcdf(

Enter Lower : 0.482

Upper : 0.580

Mean : 0.5

Standard deviation : 0.083333

Thus we get: