A test of H0: p = 0.5 against Ha: p > 0.5 uses a random sample of… Step by Step

Step by step

The question

Question: in a sample of 500, 270 say yes: at the 5% level, is the true proportion more than 0.5?

p > 0.5? 270 of 500 say yes, 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 0.5.

H₀ : p = 0.5

Hₐ is what we want to show.

Here: the proportion is more than 0.5.

H_a : p > 0.5

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.

500 times 0.5 is 250.

Both are big enough.

n × p₀ = 500 × 0.5 = 250, n × q₀ = 500 × 0.5 = 250

3 The test number

p prime, the sample's part: 270 divided by 500 is 0.54.

p′ = 270500 = 0.54

The SD uses p zero from H zero: 0.5 times 0.5, over 500, then the square root.

About 0.0224.

SD = √p₀ q₀n = √0.5 × 0.5500 ≈ 0.0224

Careful!

We act as if H zero is true.

So the SD uses 0.5, not 0.54.

Careful: in the SD: p₀ = 0.5, not p′

= 0.54

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

0.54 minus 0.5 is 0.04.

Divided by 0.0224: about 1.79.

z = p′ − p₀SD = 0.54 − 0.50.0224 ≈ 1.79

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: 0.5 in the middle, our sample at z = 1.79.

We shade the right end, from our z up.

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

Row 1.7, column .09.

For z = 1.79: the row 1.7 and the column .09 meet at 0.9633.

The right end is 1 minus 0.9633: 0.0367.

p-value = 1 − 0.9633 = 0.0367

5 Decide

The p-value, 0.0367, is less than alpha, 0.05.

Our sample would be rare if H zero were true.

So we reject H zero.

p-value 0.0367 < 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 more than 0.5.

the proportion is more than 0.5.

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 1.79: past the cut-off.

The same decision.

So it is right!

z = 1.79 > 1.645: past the cut-off, reject

8 Pick the choice

Our answer is reject H₀.

That is choice D.

D) Reject H0

9 Why not the others

A is Do not reject H0.

Wrong: the p-value 0.0367 is less than 0.05.

A) Do not reject H0 ✗ the p-value 0.0367 is less than 0.05

B is Accept Ha because 270 is more than 250.

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

B) Accept Ha because 270 is more than 250 ✗ a sample above half is not enough: the test decides

C is Accept H0.

Wrong: a test never shows H0 is true.

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

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