Traditional Method One Proportion

A researcher claims that the majority of the population supports a proposition raising taxes to help fund education. In a survey of 100 people, 52 said they supported the proposition. Use the Traditional method with α=.05 to evaluate the researcher’s claim.

If you want to try this problem on your own and just check your results, click on Leonardo to the right. Otherwise, click away from him and we’ll work through this together.

Set-up This is a test about 1 proportion, the proportion of voters who support the proposition. Here’s what we know: Population p= ? This is what the hypotheses will be about!

Set-up This is a test about 1 proportion, the proportion of voters who support the proposition. 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 majority of the population supports the proposition. That is: That’s p!

p >.5 Step 1, continued

Step 1 The Null Hypothesis has to have an equals sign, since the Null always claims there is no difference between things. The Null Hypothesis has to use the same number that shows up in the Alternate.

Step (*) Draw the picture and label the area in the critical region.

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

Remember: n = 100 p =.5 q = 1 - p = =.5 To check for normality:

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 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.5 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 raditional Method T is op-down T

Step (*): continued Once you’ve drawn the picture, start at the Top level and label the area in the critical region. Top level: Area Middle Level: Standard Units (z) 0 Bottom level: Actual Values.5

Step (*): continued Once you’ve drawn the picture, start at the Top level and label the area in the critical region. Top level: Area Middle Level: Standard Units (z) 0 Bottom level: Actual Values.05

Step 2: Move down to the middle level. Label the critical value, which is the boundary between the critical and non-critical regions. Standard Units (z) 0 Actual Values.5.05 Middle Level Put critical value here!

To find the critical value, we can use either or Click on the name of the table you would like to use.

Our picture looks like this: (we know the area to the right of the critical value, and want to know the critical value.) ? To use Table E, we want to have our picture match this one, where we know the area to the left of the critical value. Subtract the area in the right tail from the total area (1) to get the area to the left =.95

Now we can look up.9500 in the area part of Table E. Let’s zoom in!

The two areas closest to.9500 are.9495 and They are exactly the same distance from.9500; when this happens, we pick the bigger area,.9505.

The z-value associated with the area.9505 is 1.65.

Finishing up step 2: Standard Units (z) 0.05 Put critical value here! Actual Values.5

Finishing up step 2: Add the critical value to the picture. Standard Units (z) 0.05 Put critical value here! Actual Values

Step 3: Standard Units (z) 0 Actual Values Bottom level

Step 3: Standard Units (z) 0 Actual Values Bottom level Our observed value is.52. That’s clearly bigger than.5, but should it go here or here?

We want to compare our observed value,.52, to our critical value, But 1.65 is in standard units, and.52 isn’t. So we convert.52 to standard units in order to make the comparison possible.

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

=.4

Standard units (z) 0 Actual values < 1.65, so falls to the left of Line up the observed value with the test value; note that it is in the critical region..52

Standard units (z) 0 Actual value

Hooray! I hate being rejected! Most of my patients do.

Step 5: Answer the question. There is not enough evidence to support the claim that a majority of the population supports the proposition.

Let’s see this one more time!

Each click will take you to the next step; step (*) is broken into two clicks. Step 1: Step (*) Standard units (z) 0 Actual values.5.05 Step Step Step 5: There’s not enough evidence to support the claim.

And there was much rejoicing!

That’s all! Press the escape key to exit the slide show. Don’t hit the mouse or press the space bar.

Since this is a one-tailed test, look for α =.05 in this row.

Be sure to go all the way to the bottom row of Table F; this is the only row that gives us z-values! z = 1.645

Finishing up step 2: Standard Units (z) 0 Actual Units.5.05 Put critical value here!

Finishing up step 2: Add the critical value to the picture. Standard Units (z) 0.05 Put critical value here! Actual Values

Step 3: Standard Units (z) 0 Actual Values Bottom level

Step 3: Standard Units (z) 0 Actual Values Bottom level Our observed value is.52. That’s clearly bigger than.5, but should it go here or here?

We want to compare our observed value,.52, to our critical value, But is in standard units, and.52 isn’t. So we convert.52 to standard units in order to make the comparison possible.

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

=.4

Standard units (z) 0 Actual values < 1.645, so falls to the left of Line up the observed value with the test value; note that it is in the critical region..52

Standard units (z) 0 Actual value

Hooray! I hate being rejected! Most of my patients do.

Step 5: Answer the question. There is not enough evidence to support the claim that a majority of the population supports the proposition.

Let’s see this one more time!

Each click will take you to the next step; step (*) is broken into two clicks. Step 1: Step (*) Standard units (z) 0 Actual values.5.05 Step Step Step 5: There’s not enough evidence to support the claim.

And there was much rejoicing!