In a random sample of 400 voters, 230 favour a plan. A test of H0: p… Step by Step

Step by step

The question

Question: in a sample of 400, 230 say yes: at the 1% level, is the true proportion different from 0.5?

p ≠ 0.5? 230 of 400 say yes, α = 0.01

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Write the hypotheses

H₀ is the "nothing new" idea: it has the equal sign.

Here: the proportion is 0.5.

H₀ : p = 0.5

Hₐ is what we want to show.

Here: the proportion is different from 0.5.

H_a : p ≠ 0.5

Different from means too big or too small: both ends of the bell count.

Two tails.

two tails: far on either side counts

2 Big enough for the bell

The bell works if both counts are at least 5.

400 times 0.5 is 200.

Both are big enough.

n × p₀ = 400 × 0.5 = 200, n × q₀ = 400 × 0.5 = 200

3 The test number

p prime, the sample's part: 230 divided by 400 is 0.575.

p′ = 230400 = 0.575

The SD uses p zero from H zero: 0.5 times 0.5, over 400, then the square root.

About 0.025.

SD = √p₀ q₀n = √0.5 × 0.5400 ≈ 0.025

Careful!

We act as if H zero is true.

So the SD uses 0.5, not 0.575.

Careful: in the SD: p₀ = 0.5, not p′

= 0.575

z is p prime minus p zero, over the SD.

0.575 minus 0.5 is 0.075.

Divided by 0.025: about 3.

z = p′ − p₀SD = 0.575 − 0.50.025 ≈ 3

4 The p-value

The p-value is the chance of getting a sample at least this far out, if H zero is true.

p-value: the chance of a sample this far out, if H₀ is true

On the bell of H zero: 0.5 in the middle, our sample at z = 3.

Two tails: we shade both ends, as far out as our sample or more.

The z-table gives the area to the left of z.

Row 3.0, column .00.

For z = 3: the row 3.0 and the column .00 meet at 0.9987.

One end is 1 minus 0.9987: 0.0013.

Two ends: 2 times that is 0.0026.

p-value = 2 × (1 − 0.9987) = 2 × 0.0013 = 0.0026

5 Decide

The p-value, 0.0026, is less than alpha, 0.01.

Our sample would be rare if H zero were true.

So we reject H zero.

p-value 0.0026 < α = 0.01 → reject H₀

6 Say it in words

At the 1% level, there is enough evidence that

In words: at the 1% level, there is enough evidence that the proportion is different from 0.5.

the proportion is different from 0.5.

7 Check

Let's check the other way, with the cut-off.

0.01 split into two ends gives the cut-offs plus and minus 2.576, from the table, as in the cut-offs lesson.

0.01 in two ends: cut-offs ±2.576

z is 3: past the cut-off.

The same decision.

So it is right!

z = |3| > 2.576: past the cut-off, reject

8 Pick the choice

Our answer is reject H₀.

That is choice D.

D) Reject H0

9 Why not the others

A is Do not reject H0.

Wrong: the p-value 0.0026 is less than 0.01.

A) Do not reject H0 ✗ the p-value 0.0026 is less than 0.01

B is Accept Ha only at α = 0.05, not at α = 0.01.

Wrong: the p-value 0.0026 is below both.

B) Accept Ha only at α = 0.05, not at α = 0.01 ✗ the p-value 0.0026 is below both

C is Accept H0.

Wrong: a test never shows H0 is true.

C) Accept H0 ✗ a test never shows H0 is true

Let's look back.

First: write the hypotheses.

Then: big enough for the bell.

Then: the test number.

Then: the p-value.

Then: decide.

Then: say it in words.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer