Question

A poll shows that 50% of students play sports

A random sample of 20 students showed that

70% of them play sports

A number generator was used to simulate 20

samples from a population in which 50% of

students play sports. The results are shown in

the dot plot

0

0.1 0.2 03 04 05 06 07 08 0.9

1

What is the chance of getting a sample

proportion of 70% or greater?

0

0.2

03

Answer 1

The chance of getting a sample proportion of 70% or greater is 0.026.

Explanation:

We are given that a poll shows that 50% of students play sports .

A random sample of 20 students showed that 70% of them play sports.

Let
`\hat p` = sample proportion of students who play sports

The z-score probability distribution for the 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 of students who play sports = 70%

p = population proportion of students who play sports = 50%

n = sample of students = 20

Now, the chance of getting a sample proportion of 70% or greater is given by = P(
`\hat p`
`\geq` 70%)

P(
`\hat p`
`\geq` 70%) = P(
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }`
`\geq`
`\frac{0.70-0.50}{\sqrt{(0.70(1-0.70))/(20) } }` ) = P(Z
`\geq` 1.95) = 1 - P(Z < 1.95)

= 1 - 0.97441 = 0.026

The above probability is calculated by looking at the value of x = 1.95 in the z-table which has an area of 0.97441.

Hence, the chance of getting a sample proportion of 70% or greater is 0.026.