In a random sample of 220 households, 66 have a pet. What is a 90%… Step by Step

Step by step

The question

Question: in a sample of 220, 66 have a pet: find the 90% confidence interval for the true proportion.

90% interval for p: 66 of 220 have a pet

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

66 divided by 220 is 0.3.

p′ = 66220 = 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 66.

Careful: p′ is a part of 1: 66 ÷ 220, not 66

Both counts, 66 and 154, are at least 5.

So the normal bell works here.

n × p′ = 66

n × q′ = 154: 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 220: about 0.00095455.

p′ q′n = 0.3 × 0.7220 = 0.21220 ≈ 0.00095455

The square root of that is about 0.0309.

√p′ q′n = √0.00095455 ≈ 0.0309

Times 1.645: about 0.0508.

EBP = 1.645 × 0.0309 ≈ 0.0508

4 The interval

Take the error bound away from the middle: 0.3 minus 0.0508 is 0.2492.

p′ − EBP = 0.3 − 0.0508 = 0.2492

And add it: 0.3 plus 0.0508 is 0.3508.

p′ + EBP = 0.3 + 0.0508 = 0.3508

The 90 percent confidence interval is (0.2492, 0.3508).

(0.2492, 0.3508)

5 Say it in words

In words: we are 90 percent confident that the true proportion is between 24.92% and 35.08%.

90% confident: between 24.92% and 35.08%

6 Check

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

0.2492 plus 0.3508, divided by 2, is 0.3.

middle: (0.2492 + 0.3508) ÷ 2 = 0.3

Half the width should be the error bound: 0.3508 minus 0.2492, divided by 2, is 0.0508.

So it is right!

half the width: (0.3508 − 0.2492) ÷ 2 = 0.0508

7 Pick the choice

Our answer is (0.2492, 0.3508).

That is choice A.

A) (0.2492, 0.3508)

8 Why not the others

B is (0.2394, 0.3606).

Wrong: used 1.96: that is for 95%.

B) (0.2394, 0.3606) ✗ used 1.96: that is for 95%

C is (0.2691, 0.3309).

Wrong: used 1 SE: 90% needs 1.645 SE.

C) (0.2691, 0.3309) ✗ used 1 SE: 90% needs 1.645 SE

D is (0.2984, 0.3016).

Wrong: forgot the square root in the SE.

D) (0.2984, 0.3016) ✗ forgot the square root in the SE

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