P-value method

One Proportion

The problemThe mayor of Pleasantville has just signed a contract allowing a biohazards company to build a waste disposal site on

what used to be the city’s one and only park.

The mayor’s main opponent in the upcoming election claims that 90% of the town is angry over the mayor’s decision. In a survey of 50 residents, 44 responded that they were angry.

Evaluate the claim using the P-value method with α=.01.

Option to work alone and check answerIf you want to try this problem on your own and just check your answer, click on the teacher to the right. Otherwise, click away from her and we’ll work through this together.

Set-up

This test is about one proportion, the proportion of people in the town who are angry about the mayor’s decision.

Here’s what we know.

Populationp = ?

This is what the hypotheses will be about!

Set-up: sample dataSet-upThis test is about one proportion, the proportion of people in the town who are angry about the mayor’s decision.

Here’s what we know.

Populationp = ?

Samplen = 50

Step 1:State the hypotheses and identify the claim.

We are asked to evaluate the claim that the 90% of the town’s residents are angry at the mayor. That is:

proportion who are angry = .9That’s p!

Remember to convert 90% to a decimal!

The Null Hypothesisp = .9

I declare this to be the Null Hypothesis! It has an equals sign in it!

The Alternate HypothesisStep 1𝐻0 : 𝑝=.9 (𝑐𝑙𝑎𝑖𝑚)

Whenever the Null Hypothesis is the claim, we get the Alternate Hypothesis by simply negating it.

Step (*)

Draw the picture and mark off the observed value.

Not so fast!

The need to check for normalityDo we know we have a normal distribution?

With proportions, we have to investigate!

Verifying normalityRemember: n = 50

p = .9

Be sure to use the hypothesized value of p, not q = 1 - p

= 1 - .9 =.1

To check for normality:

np = (50)(.9) = 45 nq = (50)(.1) = 50

Drawing the picture: top and middle levelsStep (*) • Since we have a normal distribution, draw the picture

Top level: Area

Middle Level: Standard Units (z)

We always use z-values when working with proportions.

Drawing the picture: marking the center in standard unitsStep (*) • Since we have a normal

distribution, draw the picture

Top level: Area

Middle Level: Standard Units (z) 0

The center is always 0 in standard units.

Drawing the picture: adding the bottom levelStep (*) • Since we have a normal distribution, draw the picture

Top level: Area

Middle Level: Standard Units (z) 0

Bottom Level: Actual valuesThere are no units for proportions.

Drawing the picture: marking the center in actual values

Step (*) • Since we have a normal distribution, draw the picture

Top level: Area

Middle Level: Standard Units (z) 0

Bottom Level: Actual values .9

The number from the Null Hypothesis always goes in the center of the bottom level; that’s because we’re drawing the picture as if the Null is true.

Reminder to work bottom-upThen remember:

The -value MethodP

is ottom-upb

Adding the observed value to the picture

Step (*)

Standard Units (z) 0

Actual values .9

• Once we have the normal distribution, start at the Bottom level and mark off the observed value, .88.

Bottom level .88

.88 < .9 so it goes on the left side

Drawing in the tails

Step (*)

Standard Units (z) 0

Actual values .9.88

• Once we have the normal distribution, start at the Bottom level and mark off the observed value, .88.

Draw in the tails associated with the observed value. .88 serves as the boundary for the left tail. Since the Alternate has a sign in it, this is a two-tailed test. The right tail should be exactly the same size as the left.

Step (2): Move up to the middle level. Convert the observed value to standard units and mark this off.

Standard Units (z) 0

Actual values .9.88

Middle level:

The observed value converted to standard units is called the test value. It goes here.

Calculating the test valuez =

This is the formula for the standard error in the distribution of sample proportions.

Calculating the test value, slide 2z =

¿.88− .9

√ ( .9 )(.1)50

= .471…

≈− .47

Adding the test value to the pictureLet’s add it to our picture!

Standard Units (z) 0

Actual values .9.88

Middle level: --.43-.47 .47

Note that the boundary of the right tail is .47; marking this off is optional.

Step 3Step 3: Move up to the top level and find the area in the tails. The total area is our P-value.

Standard Units (z) 0

Actual values .9.88

-.47 .47

Top level: Area

Choice: Table E or the calculatorWe have two options for finding the P-value: we can use Table E or the calculator.

Table ECalculator

Click on the option you prefer.

Table EWe can use the left side of Table E to find the area in the left tail.

Let’s zoom in!

Finding the area in the left tail

Note: the picture above omits a number of rows from Table E so that the remaining rows can be shown on a larger scale.

.3192

Adding the area to the picture• The area in the left tail is .3192.• This means the area in the right tail is also .3192.• P = total area in both tails = 2(.3192) = .6384

Standard Units (z) 0

Actual values .9.88

-.47 .47

Top level: Area

.3192 .3192

Step 4: Decide whether or not to reject

I say we reject the sucker!

Lighten up!Let’s not reject it too hastily.

You know I hate making decisions!

Guidelines for making the decision• Compare P to α.• P = total area in two tails

= probability of getting .88 (or a value further from .9) if the Null is true

• α= maximum allowable probability of making a Type I error (rejecting the Null if it is true)

Comparing P to αP α

.6384 .01>

• The probability we would get the result we did (if the Null is true) is bigger than α.

The decisionDon’t reject the Null.

Told ya!

Step 5: Answer the question.• Talk about the claim.• Since the claim is the Null

Hypothesis, continue using the language of “rejection”.

• We did not reject the Null, so we do not reject the claim.There is not enough evidence to reject the claim that 90% of the population is angry at the mayor.

Request for a review

Could we see that one more time?

Summary

Standard units (z) 0

Actual value .9

Each click will give you one step. Step (*) is broken into two clicks.

Step 1:𝐻0 :𝜇=.9(𝑐𝑙𝑎𝑖𝑚)𝐻1 :𝜇≠ .9

Step (*).88

Step 2-.47 .47

Step 3.3192 .3192

Step 4: Don’t reject the Null.

Step 5: There is not enough evidence to reject the claim.

CelebrationAnd there was much rejoicing.

Exit the slideshowPress the escape key to exit the slide show.

If you keep clicking through the presentation, you’ll view the slides where we calculate the P-value using the calculator instead of Table E.

Finding P with the calculatorWith the calculator, there’s no need to round the critical value, so be sure you’ve still got the calculated critical value displayed on your screen.

Then hit the “shift” key followed by the “3” key.

The calculator menu

You’ll see this menu.

LEFTMIDDLE

RIGHT

Selecting from the calculator menu

Since our test value was negative, it is the boundary of the left tail. So type “1” to get the area to its left, which will be the area in the left tail.

LEFT

Entering the test value into the calculatorYou’ll see

P(

at the top of your screen. If you remembered not to delete the critical value, you’ll see it on the screen just below this. Hit the “Ans” key to enter this as the z-value.

You should get .31868.

Adding the area we got from the calculator to the picture• The area in the left tail is .31868.

• This means the area in the right tail is also .31868.• P = total area in both tails = 2(.31868) = .63736

Standard Units (z) 0

Actual values .9.88

-.47… .47…

Top level: Area

.31868 .31868

Step 4: Decide whether or not to reject (after getting P using the calculator)

Step 4: Decide whether or not to reject

I say we reject the sucker!

Lighten up!Let’s not reject it too hastily.

You know I hate making decisions!

Guidelines for making the decision(after using the calculator to get P-value)

• Compare P to α.• P = total area in two tails

= probability of getting .88 (or a value further from .9) if the Null is true

• α= maximum allowable probability of making a Type I error (rejecting the Null if it is true)

Comparing P to α (using the P-value from the calculator)

P α

.63736 .01>

• The probability we would get the result we did (if the Null is true) is bigger than α.

The decision (after using the calculator to get P)

Don’t reject the Null.

Told ya!

Step 5: Answer the question. (after using the calculator to get P)

Step 5: Answer the question.

• Talk about the claim.• Since the claim is the Null

Hypothesis, continue using the language of “rejection”.

• We did not reject the Null, so we do not reject the claim.There is not enough evidence to reject the claim that 90% of the population is angry at the mayor.

Request for a review (using calcultor to get P-value)

Could we see that one more time?

Summary (using the P-value we get from the calculator)

Standard units (z) 0

Actual value .9

Each click will give you one step. Step (*) is broken into two clicks.

Step 1:𝐻0 :𝜇=.9(𝑐𝑙𝑎𝑖𝑚)𝐻1 :𝜇≠ .9

Step (*).88

Step 2-.47… .47…

Step 3.31868 .31868

Step 4: Don’t reject the Null.

Step 5: There is not enough evidence to reject the claim.

Celebration (after using the calculator to get the P-value)And there was much rejoicing.