Test p1 != p2: 256 of 400 and 216 of 400, alpha 0.01 Step by Step

Step by step

The question

Question: in group 1, 256 of 400 say yes, and in group 2, 216 of 400: at the 1% level, is the first proportion different from the second?

p₁ ≠ p₂? 256 of 400, 216 of 400, α = 0.01

1 Write the hypotheses

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

Here: the two proportions are the same, so the difference is 0.

H₀ : p₁ = p₂

Hₐ is what we want to show.

Here: the first proportion is different from the second.

H_a : p₁ ≠ p₂

Different from: two tails.

two tails

2 The two parts

Group 1: 256/400 is 0.64.

p′₁ = 256400 = 0.64

Group 2: 216/400 is 0.54.

p′₂ = 216400 = 0.54

3 Pool the groups

If H zero is true, the two groups are the same.

So we put them together: the pooled proportion.

if H₀ is true, the groups are the same: put them together

All the yeses over everyone: 472/800 is 0.59.

p_c = 256 + 216400 + 400 = 472800 = 0.59

Careful!

We pool the counts, not the two proportions.

Careful: pool the counts: (x₁ + x₂) ÷ (n₁ + n₂)

And q c is 1 minus 0.59: 0.41.

q_c = 1 − 0.59 = 0.41

4 Standard error

The standard error, the SD of the difference: the square root of p c times q c, times 1/n 1 plus 1/n 2.

SE = √p_c × q_c × (1n₁ + 1n₂)

0.59 times 0.41 is 0.2419.

1/400 plus 1/400 is 0.005.

Multiply, then the square root: about 0.0348.

SE = √0.59 × 0.41 × (1400 + 1400) ≈ 0.0348

5 The test number

z is the difference over the SE.

0.64 minus 0.54 is 0.1.

Divided by 0.0348: about 2.88.

z = p′₁ − p′₂SE = 0.64 − 0.540.0348 ≈ 2.88

6 The p-value

The p-value is the chance of getting a sample 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 in the middle, our sample at z = 2.88.

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 2.8, column .08.

For z = 2.88: the row 2.8 and the column .08 meet at 0.9980.

One end is 1 minus 0.9980: 0.0020.

Two ends: 2 times that is 0.0040.

p-value = 2 × (1 − 0.9980) = 2 × 0.0020 = 0.0040

7 Decide

The p-value, 0.0040, is less than alpha, 0.01.

Our sample would be rare if H zero were true.

So we reject H zero.

= p-value 0.0040 < α = 0.01 → reject H₀

8 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 first proportion is different from the second.

the first proportion is different from the second.

9 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.

0.01 in two ends: cut-offs ±2.576

z is 2.88: past the cut-off.

The same decision.

So it is right!

z = |2.88| > 2.576: past the cut-off, reject

Let's look back.

First: write the hypotheses.

Then: the two parts.

Then: pool the groups.

Then: standard error.

Then: the test number.

Then: the p-value.

Then: decide.

Then: say it in words.

Last: check.

Answer