Test p1 < p2: 41 of 250 and 63 of 260, alpha 0.05 Step by Step

Step by step

The question

Question: in group 1, 41 of 250 say yes, and in group 2, 63 of 260: at the 5% level, is the first proportion less than the second?

p₁ < p₂? 41 of 250, 63 of 260, 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 less than the second.

H_a : p₁ < p₂

Less than: one tail, the left end.

one tail: the left end

2 The two parts

Group 1: 41/250 is 0.164.

p′₁ = 41250 = 0.164

Group 2: 63/260 is 0.2423.

p′₂ = 63260 = 0.2423

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: 104/510 is 0.2039.

p_c = 41 + 63250 + 260 = 104510 = 0.2039

Careful!

We pool the counts, not the two proportions.

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

And q c is 1 minus 0.2039: 0.7961.

q_c = 1 − 0.2039 = 0.7961

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)

0.2039 times 0.7961 is 0.1623.

1/250 plus 1/260 is 0.0078.

Multiply, then the square root: about 0.0357.

SE = √0.2039 × 0.7961 × (1250 + 1260) ≈ 0.0357

5 The test number

z is the difference over the SE.

0.164 minus 0.2423 is −0.0783.

Divided by 0.0357: about −2.19.

z = p′₁ − p′₂SE = 0.164 − 0.24230.0357 ≈ −2.19

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 = −2.19.

We shade the left end, from our z down.

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

Row −2.1, column .09.

For z = −2.19: the row −2.1 and the column .09 meet at 0.0143.

The left end is the table number itself: 0.0143.

p-value = 0.0143

7 Decide

The p-value, 0.0143, is less than alpha, 0.05.

Our sample would be rare if H zero were true.

So we reject H zero.

= p-value 0.0143 < 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 less than the second.

the first proportion is less than the second.

9 Check

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

0.05 in the left end gives the cut-off −1.645.

0.05 in one end: cut-off −1.645

z is −2.19: past the cut-off.

The same decision.

So it is right!

z = −2.19 < −1.645: 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