P-value method One Proportion

The 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.

If 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. Population p = ? This is what the hypotheses will be about!

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. Population p = ?

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 =.9 That’s p! Remember to convert 90% to a decimal!

p =.9 I declare this to be the Null Hypothesis! It has an equals sign in it!

Step 1 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.

Do we know we have a normal distribution? With proportions, we have to investigate!

Remember: n = 50 p =.9 q = 1 - p = =.1 To check for normality:

Step (*) 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.

Step (*) 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.

Step (*) Since we have a normal distribution, draw the picture Top level: Area Middle Level: Standard Units (z)0 Bottom Level: Actual values There are no units for proportions.

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.

Then remember: The -value Method P is ottom-up b

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 <.9 so it goes on the left side

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.

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.

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

Let’s add it to our picture! Standard Units (z) 0 Actual values.9.88 Middle level: Note that the boundary of the right tail is.47; marking this off is optional.

Step 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 Top level: Area

We have two options for finding the P-value: we can use Table E or the calculator. Table E Calculator Click on the option you prefer.

We can use the left side of Table E to find the area in the left tail. Let’s zoom in!

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

The area in the left tail is This means the area in the right tail is also P = total area in both tails = 2(.3192) =.6384 Standard Units (z) 0 Actual values Top level: Area.3192

I say we reject the sucker! Lighten up! Let’s not reject it too hastily. You know I hate making decisions!

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

Don’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.

Could we see that one more time?

Standard units (z) 0 Actual value.9 Each click will give you one step. Step (*) is broken into two clicks. Step 1: Step (*).88 Step Step Step 4: Don’t reject the Null. Step 5: There is not enough evidence to reject the claim.

And there was much rejoicing.

Press 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.

With 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.

You’ll see this menu. LEFT MIDDLE RIGHT

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

You’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

The area in the left tail is This means the area in the right tail is also P = total area in both tails = 2(.31868) = Standard Units (z) 0 Actual values ….47… Top level: Area.31868

I say we reject the sucker! Lighten up! Let’s not reject it too hastily. You know I hate making decisions!

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

Don’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.

Could we see that one more time?

Standard units (z) 0 Actual value.9 Each click will give you one step. Step (*) is broken into two clicks. Step 1: Step (*).88 Step ….47… Step Step 4: Don’t reject the Null. Step 5: There is not enough evidence to reject the claim.

And there was much rejoicing.