In a random sample of 500 shoppers, 150 used a coupon. What is a 95%… Step by Step

Step by step

The question

Question: in a sample of 500, 150 say yes: find the 95% confidence interval for the true proportion.

95% interval for p: 150 of 500 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

150 divided by 500 is 0.3.

p′ = 150500 = 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 150.

Careful: p′ is a part of 1: 150 ÷ 500, not 150

Both counts, 150 and 350, are at least 5.

So the normal bell works here.

n × p′ = 150

n × q′ = 350: both at least 5

2 Find z*

95 percent of the bell is in the middle.

That leaves 5 percent: 2.5 percent in each end.

95% in the middle: 5% left over, 2.5% in each end

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

Left of z* are the middle and one end: 0.95 plus 0.025 is 0.9750.

left of z* = 0.95 + 0.025 = 0.9750

Now we look for 0.9750 inside the table.

0.9750 is in the table: row 1.9, column .06.

So z = 1.96.

z* = 1.96

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 500: 0.00042.

p′ q′n = 0.3 × 0.7500 = 0.21500 = 0.00042

The square root of that is about 0.0205.

√p′ q′n = √0.00042 ≈ 0.0205

Times 1.96: about 0.0402.

EBP = 1.96 × 0.0205 ≈ 0.0402

4 The interval

Take the error bound away from the middle: 0.3 minus 0.0402 is 0.2598.

p′ − EBP = 0.3 − 0.0402 = 0.2598

And add it: 0.3 plus 0.0402 is 0.3402.

p′ + EBP = 0.3 + 0.0402 = 0.3402

The 95 percent confidence interval is (0.2598, 0.3402).

(0.2598, 0.3402)

5 Say it in words

In words: we are 95 percent confident that the true proportion is between 25.98% and 34.02%.

95% confident: between 25.98% and 34.02%

6 Check

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

0.2598 plus 0.3402, divided by 2, is 0.3.

middle: (0.2598 + 0.3402) ÷ 2 = 0.3

Half the width should be the error bound: 0.3402 minus 0.2598, divided by 2, is 0.0402.

So it is right!

half the width: (0.3402 − 0.2598) ÷ 2 = 0.0402

7 Pick the choice

Our answer is (0.2598, 0.3402).

That is choice B.

B) (0.2598, 0.3402)

8 Why not the others

A is (0.2663, 0.3337).

Wrong: used 1.645: that is for 90%.

A) (0.2663, 0.3337) ✗ used 1.645: that is for 90%

C is (0.2795, 0.3205).

Wrong: used 1 SE: 95% needs 1.96 SE.

C) (0.2795, 0.3205) ✗ used 1 SE: 95% needs 1.96 SE

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