A town says 60% of its residents support a new park. In a random… Step by Step

Step by step

The question

Question: in a sample of 200, 105 agree: at the 5% level, is the true proportion less than 0.6?

p < 0.6? 105 of 200 agree, 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 proportion is 0.6.

H₀ : p = 0.6

Hₐ is what we want to show.

Here: the proportion is less than 0.6.

H_a : p < 0.6

Less than means only the left end counts.

One tail.

one tail: the left end

2 Big enough for the bell

The bell works if both counts are at least 5.

200 times 0.6 is 120.

200 times 0.4 is 80.

Both are big enough.

n × p₀ = 200 × 0.6 = 120, n × q₀ = 200 × 0.4 = 80

3 The test number

p prime, the sample's part: 105 divided by 200 is 0.525.

p′ = 105200 = 0.525

The SD uses p zero from H zero: 0.6 times 0.4, over 200, then the square root.

About 0.0346.

SD = √p₀ q₀n = √0.6 × 0.4200 ≈ 0.0346

Careful!

We act as if H zero is true.

So the SD uses 0.6, not 0.525.

Careful: in the SD: p₀ = 0.6, not p′

= 0.525

z is p prime minus p zero, over the SD.

0.525 minus 0.6 is −0.075.

Divided by 0.0346: about −2.17.

z = p′ − p₀SD = 0.525 − 0.60.0346 ≈ −2.17

4 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: 0.6 in the middle, our sample at z = −2.17.

We shade the left end, from our z down.

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

Row −2.1, column .07.

For z = −2.17: the row −2.1 and the column .07 meet at 0.0150.

The left end is the table number itself: 0.0150.

p-value = 0.0150

5 Decide

The p-value, 0.0150, is less than alpha, 0.05.

Our sample would be rare if H zero were true.

So we reject H zero.

p-value 0.0150 < α = 0.05 → reject H₀

6 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 proportion is less than 0.6.

the proportion is less than 0.6.

7 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 −2.17: past the cut-off.

The same decision.

So it is right!

z = −2.17 < −1.645: past the cut-off, reject

8 Pick the choice

Our answer is reject H₀.

That is choice A.

A) Reject H0: there is convincing evidence that fewer than 60% support the park

9 Why not the others

B is Do not reject H0.

Wrong: the p-value 0.015 is less than 0.05.

B) Do not reject H0 ✗ the p-value 0.015 is less than 0.05

C is Accept H0: 60% support the park.

Wrong: a test never shows H0 is true.

C) Accept H0: 60% support the park ✗ a test never shows H0 is true

D is Accept Ha because 105 is more than half of 200.

Wrong: the claim is about 60%, not about half.

D) Accept Ha because 105 is more than half of 200 ✗ the claim is about 60%, not about half

Let's look back.

First: write the hypotheses.

Then: big enough for the bell.

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