In a random sample of 160 city drivers, 32 had an accident last… Step by Step

Step by step

The question

Question: in group 1, 32 of 160 say yes, and in group 2, 48 of 160: at the 5% level, is the first proportion less than the second?

p₁ < p₂? 32 of 160, 48 of 160, α = 0.05

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 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: 32/160 is 0.2.

p′₁ = 32160 = 0.2

Group 2: 48/160 is 0.3.

p′₂ = 48160 = 0.3

3 Pool the groups

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

So we put them together: the pooled proportion, p c, with c for combined.

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

All the yeses over everyone: 80/320 is 0.25.

p_c = 32 + 48160 + 160 = 80320 = 0.25

Careful!

We pool the counts, not the two proportions.

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

And q c is 1 minus 0.25: 0.75.

q_c = 1 − 0.25 = 0.75

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.

Each group has its own spread, and for a difference the two spreads add.

SE = √p_c × q_c × (1n₁ + 1n₂)

0.25 times 0.75 is 0.1875.

1/160 plus 1/160 is 0.0125.

Multiplied: 0.002344.

Then the square root: about 0.0484.

SE = √0.25 × 0.75 × (1160 + 1160) ≈ 0.0484

5 The test number

z is the difference over the SE.

0.2 minus 0.3 is −0.1.

Divided by 0.0484: about −2.07.

z = p′₁ − p′₂SE = 0.2 − 0.30.0484 ≈ −2.07

6 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 in the middle, our sample at z = −2.07.

We shade the left end, from our z down.

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

Row −2.0, column .07.

For z = −2.07: the row −2.0 and the column .07 meet at 0.0192.

The left end is the table number itself: 0.0192.

p-value = 0.0192

7 Decide

The p-value, 0.0192, is less than alpha, 0.05.

Our sample would be rare if H zero were true.

So we reject H zero.

p-value 0.0192 < 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, from the table, as in the cut-offs lesson.

0.05 in one end: cut-off −1.645

z is −2.07: past the cut-off.

The same decision.

So it is right!

z = −2.07 < −1.645: past the cut-off, reject

10 Pick the choice

Our answer is reject H₀.

That is choice B.

B) Reject H0

11 Why not the others

A is Do not reject H0.

Wrong: the p-value 0.0192 is less than 0.05.

A) Do not reject H0 ✗ the p-value 0.0192 is less than 0.05

C is Accept H0.

Wrong: a test never shows H0 is true.

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

D is Accept Ha because 32 is less than 48.

Wrong: the difference must be checked by the test.

D) Accept Ha because 32 is less than 48 ✗ the difference must be checked by the test

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.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer