In a random sample of 90 students, 27 bike to school. What is a 90%… Step by Step

Step by step

The question

Question: in a sample of 90, 27 say yes: find the 90% confidence interval for the true proportion.

90% interval for p: 27 of 90 say yes

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Name the numbers

p prime is the part of the sample that says yes: x divided by n.

p′ = the part of the sample: x ÷ n

27 divided by 90 is 0.3.

p′ = 2790 = 0.3

q prime is the rest, the part that says no: 1 minus 0.3 is 0.7.

q′ = 1 − 0.3 = 0.7

Careful!

p prime is the fraction 0.3, not the count 27.

Careful: p′ is a part of 1: 27 ÷ 90, not 27

Both counts, 27 and 63, are at least 5.

So the normal bell works here.

n × p′ = 27

n × q′ = 63: both at least 5

2 Find z*

90 percent of the bell is in the middle.

That leaves 10 percent: 5 percent in each end.

90% in the middle: 10% left over, 5% in each end

z* marks the cut between the middle and the ends.

Left of z* are the middle and one end: 0.9 plus 0.05 is 0.9500.

left of z* = 0.9 + 0.05 = 0.9500

Now we look for 0.9500 inside the table.

0.9500 is halfway between 0.9495 and 0.9505.

So z is halfway between 1.64 and 1.65: 1.645.

z* = 1.645

3 Error bound

The error bound for a proportion: z* times the square root of p prime times q prime over n.

That root is the SD of sample proportions, as σ over the square root of n was for means.

EBP = z* × √p′ q′n

0.3 times 0.7 is 0.21.

Divided by 90: about 0.00233333.

p′ q′n = 0.3 × 0.790 = 0.2190 ≈ 0.00233333

The square root of that is about 0.0483.

√p′ q′n = √0.00233333 ≈ 0.0483

Times 1.645: about 0.0795.

EBP = 1.645 × 0.0483 ≈ 0.0795

4 The interval

Take the error bound away from the middle: 0.3 minus 0.0795 is 0.2205.

p′ − EBP = 0.3 − 0.0795 = 0.2205

And add it: 0.3 plus 0.0795 is 0.3795.

p′ + EBP = 0.3 + 0.0795 = 0.3795

The 90 percent confidence interval is (0.2205, 0.3795).

(0.2205, 0.3795)

5 Say it in words

In words: we are 90 percent confident that the true proportion is between 22.05% and 37.95%.

90% confident: between 22.05% and 37.95%

6 Check

Let's check: the middle of the interval should be 0.3.

0.2205 plus 0.3795, divided by 2, is 0.3.

middle: (0.2205 + 0.3795) ÷ 2 = 0.3

Half the width should be the error bound: 0.3795 minus 0.2205, divided by 2, is 0.0795.

So it is right!

half the width: (0.3795 − 0.2205) ÷ 2 = 0.0795

7 Pick the choice

Our answer is (0.2205, 0.3795).

That is choice B.

B) (0.2205, 0.3795)

8 Why not the others

A is (0.2053, 0.3947).

Wrong: used 1.96: that is for 95%.

A) (0.2053, 0.3947) ✗ used 1.96: that is for 95%

C is (0.27, 0.33).

Wrong: used the margin 0.03.

C) (0.27, 0.33) ✗ used the margin 0.03

D is (0.25, 0.35).

Wrong: guessed a margin of 0.05: work it out.

D) (0.25, 0.35) ✗ guessed a margin of 0.05: work it out

Let's look back.

First: name the numbers.

Then: find z*.

Then: error bound.

Then: the interval.

Then: say it in words.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer