Test p1 != p2: 120 of 200 and 100 of 200, alpha 0.05 Step by Step

Step by step

The question

Question: in group 1, 120 of 200 say yes, and in group 2, 100 of 200: at the 5% level, is the first proportion different from the second?

p₁ ≠ p₂? 120 of 200, 100 of 200, α = 0.05

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: 120/200 is 0.6.

p′₁ = 120200 = 0.6

Group 2: 100/200 is 0.5.

p′₂ = 100200 = 0.5

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: 220/400 is 0.55.

p_c = 120 + 100200 + 200 = 220400 = 0.55

Careful!

We pool the counts, not the two proportions.

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

And q c is 1 minus 0.55: 0.45.

q_c = 1 − 0.55 = 0.45

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)

0.55 times 0.45 is 0.2475.

1/200 plus 1/200 is 0.01.

Multiply, then the square root: about 0.0498.

SE = √0.55 × 0.45 × (1200 + 1200) ≈ 0.0498

5 The test number

z is the difference over the SE.

0.6 minus 0.5 is 0.1.

Divided by 0.0498: about 2.01.

z = p′₁ − p′₂SE = 0.6 − 0.50.0498 ≈ 2.01

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 = 2.01.

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 2.0, column .01.

For z = 2.01: the row 2.0 and the column .01 meet at 0.9778.

One end is 1 minus 0.9778: 0.0222.

Two ends: 2 times that is 0.0444.

p-value = 2 × (1 − 0.9778) = 2 × 0.0222 = 0.0444

7 Decide

The p-value, 0.0444, is less than alpha, 0.05.

Our sample would be rare if H zero were true.

So we reject H zero.

= p-value 0.0444 < α = 0.05 → reject H₀

8 Say it in words

At the 5% level, there is enough evidence that

In words: at the 5% 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.

0.05 split into two ends gives the cut-offs plus and minus 1.96.

0.05 in two ends: cut-offs ±1.96

z is 2.01: past the cut-off.

The same decision.

So it is right!

z = |2.01| > 1.96: 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