Test p1 > p2: 60 of 200 and 50 of 200, alpha 0.05 Step by Step

Step by step

The question

Question: in group 1, 60 of 200 say yes, and in group 2, 50 of 200: at the 5% level, is the first proportion more than the second?

p₁ > p₂? 60 of 200, 50 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 more than the second.

H_a : p₁ > p₂

More than: one tail, the right end.

one tail: the right end

2 The two parts

Group 1: 60/200 is 0.3.

p′₁ = 60200 = 0.3

Group 2: 50/200 is 0.25.

p′₂ = 50200 = 0.25

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: 110/400 is 0.275.

p_c = 60 + 50200 + 200 = 110400 = 0.275

Careful!

We pool the counts, not the two proportions.

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

And q c is 1 minus 0.275: 0.725.

q_c = 1 − 0.275 = 0.725

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)

0.275 times 0.725 is 0.1994.

1/200 plus 1/200 is 0.01.

Multiply, then the square root: about 0.0447.

SE = √0.275 × 0.725 × (1200 + 1200) ≈ 0.0447

5 The test number

z is the difference over the SE.

0.3 minus 0.25 is 0.05.

Divided by 0.0447: about 1.12.

z = p′₁ − p′₂SE = 0.3 − 0.250.0447 ≈ 1.12

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 = 1.12.

We shade the right end, from our z up.

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

Row 1.1, column .02.

For z = 1.12: the row 1.1 and the column .02 meet at 0.8686.

The right end is 1 minus 0.8686: 0.1314.

p-value = 1 − 0.8686 = 0.1314

7 Decide

The p-value, 0.1314, is more than alpha, 0.05.

Our sample is not rare enough.

So we do not reject H zero.

= p-value 0.1314 > α = 0.05 → do not reject H₀

8 Say it in words

At the 5% level, there is not enough evidence that

In words: at the 5% level, there is not enough evidence that the first proportion is more than the second.

the first proportion is more than the second.

This does not prove H zero is true.

We just cannot rule it out.

not rejecting H₀ does not prove H₀

9 Check

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

0.05 in the right end gives the cut-off 1.645.

0.05 in one end: cut-off 1.645

z is 1.12: not past the cut-off.

The same decision.

So it is right!

z = 1.12 < 1.645: not past it, do not 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