P-value Method 2 proportions

A resident of a small housing complex has a pet monkey who likes to sit out on the porch and smoke cigarettes. Some of the neighbors object, arguing that a)monkeys are inappropriate pets, and b)the neighborhood children will be more likely to take up smoking after seeing the monkey smoke on a regular basis. Acknowledgement: This picture and all the other picture s of monkeys in this presentation were found at

Some of the residents are proposing a local ban on monkeys as pets, and the homeowner is concerned. In talking to his neighbors, he thinks he has noticed a difference in the attitudes of those who are fans of “The Big Bang Theory” and those who aren’t. (On the TV show “The Big Bang Theory” the character Amy Farrah Fowler is a neurobiologist who is studying a smoking monkey.) He decides to test his theory by conducting a survey in which he first asks people if they are fans of the show, and then asks them if they object to his pet monkey.

Of 30 neighbors who say they are fans of the show, 14 report that they object to pet monkeys in the neighborhood. Of 25 neighbors who are not fans of the show, 13 say they object to pet monkeys. Use the P-value method with α =.05 to evaluate the resident’s claim that the proportion of fans who object to monkeys as pets is different from the proportion of non- fans.

If you want to try this problem on your own and just check your answer, click on the monkeys to the right when you’re ready to check your answer. Otherwise, click away from the monkeys (or just hit the space bar) and we’ll work through it together.

Set-up Here’s the data he’s collected: Keep this as a fraction, since its decimal equivalent repeats, and we don’t want to round until after we’ve done all our calculations.

Step 1 State the Hypotheses and identify the claim. Recall that we were asked to….

Observe that there are no hats on these p’s; that’s because the hypotheses are always about the (unknown) population proportions, not the (known) sample proportions. Seeing no equals sign, we can conclude that this is the Alternate Hypothesis.

The Null hypothesis should compare the same quantities, but have an equals sign.

Let’s rewrite the hypotheses by subtracting everything over to one side. That way, the number that shows up in the hypotheses will be the same number that shows up at the center of our distribution. Good idea! And since later on we’ll have to subtract the sample proportions in the same order, let’s set it up so we’re subtracting “bigger minus smaller.” The sample proportion for non- fans was bigger in this case, so that means “non-fans minus fans.”

Note that we could have subtracted in the other order. That would mean that when we subtract the sample proportions later on, we’d get a negative number instead of a positive number.

Step (*) Draw the picture and mark off the observed value. Hold on there! Don’t go drawing a normal distribution unless you know you have one!

Hypothesizing that the two population proportions are equal is equivalent to saying that there aren’t two different proportions for these two populations, but just one proportion that works for everyone. While the two sample proportions approximate this, we’ll get a better estimate by pooling all our data.

Keep this as a fraction to avoid rounding before using it in calculations.

Step (*) Since we have a normal distribution, draw a normal curve. Top level: Area Middle level: standard units(z) We always use z-values when we are working with proportions.

Step (*): Since we have a normal distribution, draw a normal curve. Top level: Area Middle Level: Standard Units (z) 0 The center is always 0 in standard units. Label this whenever you draw the picture.

Step (*): Since we have a normal distribution, draw a normal curve. 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 a normal curve. Top level: Area Middle Level: Standard Units (z) 0 Bottom level: Actual Values 0 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 (*) continued Once you’ve drawn the picture, start at the bottom level and mark off the observed difference. Standard Units (z) 0 Actual values0 Bottom level Remember to subtract “non-fans minus fans” since that’s the order we used in the hypotheses!

Step (*) continued Once you’ve drawn the picture, start at the bottom level and mark off the observed difference. Standard Units (z) 0 Actual values0 Bottom level

Step (*) continued Once you’ve drawn the picture, start at the bottom level and mark off the observed difference. Standard Units (z) 0 Actual values0 Bottom level

Step (*) continued Once you’ve drawn the picture, start at the bottom level and mark off the observed difference. Standard Units (z) 0 Actual values0 Bottom level

Step 2 Standard Units (z) 0 Actual values0 Middle level test value goes here

Calculating the test value: standard error for a difference in two proportions that are hypothesized to be equal

Calculating the test value: This is the hypothesized difference; it is the same number that shows up in the hypotheses and marks the center of our distribution.

Calculating the test value:

Adding the test value to the picture: Standard Units (z) 0 Actual values0 Middle level test value goes here By symmetry, the boundary of the left tail is Adding this is optional, but it makes the next step a little nicer.

Step 3: Move up to the top level and find the P-value, which is the area in the tails. Standard Units (z) 0 Actual values Top level: Area

We can use either Table E or the calculator to find the P-value. Click on the table if you’d rather use a Table (specifically, Table E) to calculate the P-value. Click on the abacus if you’d prefer to use a calculator to find the P-value; we’ll actually use the Casio fx-115MS plus, which has a few more features than the abacus.

Since our standard units are z-values, we use Table E to find the area in the tails. If we use the negative z-value, -.39, we will use the left side of table E and look up the area in the left tail. Since we have 2 tails, we will double this to get our P-value. We could use the positive z- value,.39. Table E will give us the area to the left of this z- value and we would have to subtract it from 1 to get the area in the right tail. Again, we would have to double this to get the P- value.

Looks like it’ll be easier to use the left side of Table E. Let’s zoom in!

Missing rows Note: the “missing rows” have been deleted in order to make the rest of the table fit better on the screen..3483

Standard units (z) 0 Actual values The area in the left tail is By symmetry, the area in the right tail is also P = total area in both tails = 2(.3483) =.6966

Step 4: Decide whether or not to reject the Null Compare P to α.

P _ α__ > P > α Do not reject the Null.

Step 5: Answer the question. The question was about the claim. The claim is the Alternate Hypothesis, so we have to use the language of “support.” We didn’t reject the Null, so we don’t support the Alternate.

There is not enough evidence to support the claim that the proportion of “The Big Bang Theory” fans who object to monkeys as pets is different from the proportion of non-fans who object. Could we go over that again?

Each click will give you one step. Step (*) is broken into two clicks. Step 1. Step (*) Standard units (z) 0 Actual values0 Step Step Step 4: Do not reject the Null. Step 5: There is not enough evidence to support the claim.

And there was much rejoicing

Press the escape key to exit the slide show. If you continue to click through the slideshow, you’ll re-work the problem using yoru calculator to get the P- value.

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.

The calculator menu You’ll see this menu. LEFT MIDDLE RIGHT

Since our critical value is the boundary of the right tail, we’ll get the area in one tail if we select the area to the right of our critical value. Hit “3” to select this option.

You’ll see this: Press the “Ans” key to enter the critical value into this function. Then hit the “=‘’ key. You should get.34681; this is the area in the right tail.

Standard units (z) 0 Actual values0.39… -.39… The area in the right tail is By symmetry, the area in the left tail is also P = total area in both tails = 2(.34681) =.69362

Step 4: Decide whether or not to reject the Null Compare P to α.

P _ α__ > P > α Do not reject the Null.

Step 5: Answer the question! The question was about the claim. The claim is the Alternate Hypothesis, so we have to use the language of “support.” We didn’t reject the Null, so we don’t support the Alternate.

There is not enough evidence to support the claim that the proportion of “The Big Bang Theory” fans who object to monkeys as pets is different from the proportion of non-fans who object. Could we go over that again?

Each click will give you one step. Step (*) is broken into two clicks. Step 1. Step (*) Standard units (z) 0 Actual values0 Step 2.39… -.39… Step Step 4: Do not reject the Null. Step 5: There is not enough evidence to support the claim.

And there was much rejoicing!