TESTING CLAIMS ABOUT THE PROPORTIONLecture 17

TESTING CLAIMS ABOUT THE PROPORTION Last lecture was the hard part

Now you know everything you need to know about testing claims and can apply your knowledge to testing claims about other parameters besides the mean

This lecture we’ll test claims about the population proportion… but only a very small part of this process will be at all new to you

You remember the basics from when we covered estimating population proportions in Lecture #14 If a variable is binomial, we call p the proportion in the population for which the variable was

in one of the two categories

In a sample from that population, the sample proportion is called 𝑝 and equals 𝑥

𝑛, where x is

the number of times the variable was in the category of interest and n, of course, is the sample size

TESTING CLAIMS ABOUT THE PROPORTION STEPS IN TESTING CLAIMS

A. State the hypotheses and identify the claim.

B. State the values of x and n.

Perform the test using the sample data or statistics, and make a diagram showing the distribution, the parameter and the statistic.

Shade and label the p-value, which should be rounded to the nearest thousandth.

C. Decide whether or not to reject the null hypothesis.

D. Summarize the results.

You’ll note that these steps from before are perfectly applicable to testing claims about the proportion, with the addition of stating the values of x and n.

The only differences are:

That the parameter is p instead of 𝜇

The value of the parameter mentioned in the claim is 𝑝0 instead of 𝜇0

The statistic is 𝑝 instead of 𝑥

We will never use the t-distribution in testing claims about the proportion, only z

TESTING THE FIRST CLAIM

Here’s the claim: At most one-third of people stopped by the CHP for suspected DUI test above the legal limit

To test this claim, a random sample of 200 people stopped for suspected DUI is selected, and of these it turned out that 36% tested above the legal limit

Let’s choose the 5% significance level

A. Remembering to use p for the parameter, and that ‘at most’ encompasses the ‘is less than’ and the ‘equals’ options, the hypotheses become

𝐻0: 𝑝 ≤1

3(Claim)

𝐻0: 𝑝 >1

3

This is a right-tailed test

TESTING THE FIRST CLAIM

Let’s skip the build-up and go directly to the Stat Test menu on the calculator

We’ll be using 1-PropZTest

Putting 1

3in for 𝑝0 seems obvious

when we enter that the calculator uses this repeating decimal in all its glory: .3333333333…

But the next line asks for x, which is the number of people out of the 200 who tested above the

legal limit, and that’s information we weren’t given directly

We’ll have to use a little arithmetic, or a little algebra

TESTING THE FIRST CLAIM

Perhaps you realize that since 36% of the 200 people in the sample tested above the legal limit,

to find out how many people that actually is you’ll take 36% of 200 and come up with x = 72

If you don’t readily see that, you’ll have to work harder

Take the equation 𝑝 =𝑥

𝑛and substitute 0.36 for 𝑝 and 200 for n:

0.36 =𝑥

200

Multiply both sides by 200 and again you’ll get x = 72

TESTING THE FIRST CLAIM

Either way, the 1-PropZTest screen looks like this:

Upon calculation, the screen becomes:

Watch out for all the p’s

The one with the hat is the sample proportion (which we didn’t actually input – the calculator

computed it from the x and the n), and the one without the hat is our p-value

TESTING THE FIRST CLAIM

B. x = 72; n = 200

C. Here again is the way to decide whether or not to reject the null hypothesis: If 𝑝 < 𝛼, reject 𝐻0 If 𝑝 > 𝛼, don’t reject 𝐻0

In this case, 0.212 > 0.05, or 𝑝 > 𝛼, so do not reject 𝐻0

TESTING THE FIRST CLAIM

D) Here again are the rectangles for summarizing the results:

Since we didn’t reject 𝑯𝟎 and since the claim was 𝑯𝟎, we’re in the upper-right triangle

There isn’t sufficient evidence to reject the claim that at most one-third of people stopped by the CHP for suspected DUI test above the legal limit

Reject 𝑯𝟎 Do not reject 𝑯𝟎

The claim is 𝑯𝟎

There is sufficient

evidence to reject the

claim that…

There isn’t sufficient

evidence to reject the

claim that…

The claim is 𝑯𝟏

There is sufficient

evidence to support

the claim that…

There isn’t sufficient

evidence to support

the claim that…

TESTING THE SECOND CLAIM

Here’s the second claim: The majority of Mendocino College students are female. Let 𝜶 =𝟎. 𝟏𝟎. Our sample is the Class Data Base, in which 60 of the 102 people in the sample are female

What is a majority? It’s more than half

If a group is split 50-50, there isn’t a majority

Any time you see the phrase “A majority…” you don’t have to read any further to translate it to a mathematical sentence It’s always p > 0.5.

Likewise, “A minority…” always translates as p < 0.5

And of course it’s always the alternative hypothesis. The null hypothesis has to include the ‘is less than’ and ‘equals’ options

TESTING THE SECOND CLAIM

Here’s the second claim: The majority of Mendocino College students are female. Let 𝜶 =𝟎. 𝟏𝟎. Our sample is the Class Data Base, in which 60 of the 102 people in the sample are female

A. 𝐻0: 𝑝 ≤1

3

𝐻0: 𝑝 >1

3(Claim)

B. x = 60; n = 102

TESTING THE SECOND CLAIM

Here’s the second claim: The majority of Mendocino College students are female. Let 𝜶 =𝟎. 𝟏𝟎. Our sample is the Class Data Base, in which 60 of the 102 people in the sample are female

Notice that the calculator found 𝑝 for us. Here’s the diagram:

C. 0.037 < 0.10, or 𝑝 < 𝛼, so reject𝐻0

D. Since we did reject 𝐻0, and since the claim was 𝐻1, we’re in the lower-left rectangle

There is sufficient evidence to support the claim that the majority of Mendocino College students are female

ACTIVITY #17: TESTING CLAIMS ABOUT THE PROPORTION

STEPS IN TESTING CLAIMSA. State the hypotheses and identify the claim.B. State the values of x and n.

Perform the test using the sample data or statistics, and make a diagram showing the distribution, the parameter and the statistic.

Shade and label the p-value, which should be rounded to the nearest thousandth. C. Decide whether or not to reject the null hypothesis.D. Summarize the results.

Use the 5% significance level in testing these claims.

1. It has been claimed that a minority of Americans are in favor of national health insurance. In a survey in which 400 people were interviewed, it was found that 47% were in favor of national health insurance.

2. A researcher studying demographics thinks that the majority of Mendocino College students are natives of Lake or Mendocino County. Test this claim.

3. Two-thirds of Mendocino College students own at least one pet.