In a random sample of 200 voters, 84 support a plan. What is a 95%… Step by Step

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The question

Question: in a sample of 200, 84 say yes: find the 95% confidence interval for the true proportion.

95% interval for p: 84 of 200 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

84 divided by 200 is 0.42.

p′ = 84200 = 0.42

q prime is the rest, the part that says no: 1 minus 0.42 is 0.58.

q′ = 1 − 0.42 = 0.58

Careful!

p prime is the fraction 0.42, not the count 84.

Careful: p′ is a part of 1: 84 ÷ 200, not 84

Both counts, 84 and 116, are at least 5.

So the normal bell works here.

n × p′ = 84

n × q′ = 116: 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

0.42 times 0.58 is 0.2436.

Divided by 200: 0.001218.

p′ q′n = 0.42 × 0.58200 = 0.2436200 = 0.001218

The square root of that is about 0.0349.

√p′ q′n = √0.001218 ≈ 0.0349

Times 1.96: about 0.0684.

EBP = 1.96 × 0.0349 ≈ 0.0684

4 The interval

Take the error bound away from the middle: 0.42 minus 0.0684 is 0.3516.

p′ − EBP = 0.42 − 0.0684 = 0.3516

And add it: 0.42 plus 0.0684 is 0.4884.

p′ + EBP = 0.42 + 0.0684 = 0.4884

The 95 percent confidence interval is (0.3516, 0.4884).

(0.3516, 0.4884)

5 Say it in words

In words: we are 95 percent confident that the true proportion is between 35.16% and 48.84%.

95% confident: between 35.16% and 48.84%

6 Check

Let's check: the middle of the interval should be 0.42.

0.3516 plus 0.4884, divided by 2, is 0.42.

middle: (0.3516 + 0.4884) ÷ 2 = 0.42

Half the width should be the error bound: 0.4884 minus 0.3516, divided by 2, is 0.0684.

So it is right!

half the width: (0.4884 − 0.3516) ÷ 2 = 0.0684

7 Pick the choice

Our answer is (0.3516, 0.4884).

That is choice A.

A) (0.3516, 0.4884)

8 Why not the others

B is (0.3626, 0.4774).

Wrong: used 1.645: that is for 90%.

B) (0.3626, 0.4774) ✗ used 1.645: that is for 90%

C is (0.4176, 0.4224).

Wrong: forgot the square root in the SE.

C) (0.4176, 0.4224) ✗ forgot the square root in the SE

D is (0.3851, 0.4549).

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

D) (0.3851, 0.4549) ✗ used 1 SE: 95% needs 1.96 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