A test of H0: p = 0.2 against Ha: p > 0.2 uses a random sample of… Step by Step

Step by step

The question

Question: in a sample of 270, 63 own a dog: at the 5% level, is the true proportion more than 0.2?

p > 0.2? 63 of 270 own a dog, 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 0.2.

H₀ : p = 0.2

Hₐ is what we want to show.

Here: the proportion is more than 0.2.

H_a : p > 0.2

More than means only the right end counts.

One tail.

one tail: the right end

2 Big enough for the bell

The bell works if both counts are at least 5.

270 times 0.2 is 54.

270 times 0.8 is 216.

Both are big enough.

n × p₀ = 270 × 0.2 = 54, n × q₀ = 270 × 0.8 = 216

3 The test number

p prime, the sample's part: 63 divided by 270 is 0.2333.

p′ = 63270 = 0.2333

The SD uses p zero from H zero: 0.2 times 0.8, over 270, then the square root.

About 0.0243.

SD = √p₀ q₀n = √0.2 × 0.8270 ≈ 0.0243

Careful!

We act as if H zero is true.

So the SD uses 0.2, not 0.2333.

Careful: in the SD: p₀ = 0.2, not p′

= 0.2333

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

0.2333 minus 0.2 is 0.0333.

Divided by 0.0243: about 1.37.

z = p′ − p₀SD = 0.2333 − 0.20.0243 ≈ 1.37

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: 0.2 in the middle, our sample at z = 1.37.

We shade the right end, from our z up.

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

Row 1.3, column .07.

For z = 1.37: the row 1.3 and the column .07 meet at 0.9147.

The right end is 1 minus 0.9147: 0.0853.

p-value = 1 − 0.9147 = 0.0853

5 Decide

The p-value, 0.0853, is more than alpha, 0.05.

Our sample is not rare enough.

So we do not reject H zero.

p-value 0.0853 > α = 0.05 → do not reject H₀

6 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 proportion is more than 0.2.

the proportion is more than 0.2.

This does not prove H zero is true.

We just cannot rule it out.

not rejecting H₀ does not prove H₀

7 Check

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

0.05 in the right 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 1.37: not past the cut-off.

The same decision.

So it is right!

z = 1.37 < 1.645: not past it, do not reject

8 Pick the choice

Our answer is do not reject H₀.

That is choice B.

B) Do not reject H0

9 Why not the others

A is Reject H0.

Wrong: the p-value 0.0853 is more than 0.05.

A) Reject H0 ✗ the p-value 0.0853 is more than 0.05

C is Accept H0: exactly 20% own a dog.

Wrong: a test never shows H0 is true.

C) Accept H0: exactly 20% own a dog ✗ a test never shows H0 is true

D is Accept Ha because 63/270 is more than 0.2.

Wrong: a sample above 0.2 is not enough: the test decides.

D) Accept Ha because 63270 is more than 0.2 ✗ a sample above 0.2 is not enough: the test decides

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