Traditional Method One mean, sigma known

The Problem In 2004, the average monthly Social Security benefit for retired workers was $ with a standard deviation of $ In a sample of 50 retired workers in 2005, their average Social Security benefit was $ If the standard deviation has not changed, is there sufficient evidence at α=.05 to support a claim that the benefits have increased? Use the traditional method. Source: Bluman, Elemenary Statistics, eighth edition, citing the New York Times Almanac.

If you want to work through this problem on your own and just check your answer, click the person to the left when you’re ready to check your work. Otherwise, click away from the person or press the space bar, and we’ll work through this problem together.

Set-up This is a test about 1 mean, the mean in This person is confused; she thought it was about 2 means, one for 2004 and one for 2005! Click on her if you share her confusion. Otherwise, move the mouse away from her and click (or just hit the space bar) to keep going.

Here’s what we know: Population μ= ? This is what the hypotheses will be about!

Here’s what we know: Population μ= ? σ=98 Since we are told to assume the standard deviation has not changed, we can use the population standard deviation from 2004 as the population standard deviation for 2005.

Population μ= ? σ=98

Step 1: State the hypotheses and identify the claim. We are asked to evaluate the claim that benefits have increased. That is: That’s μ! That’s

So the claim is…. Do you see an equals sign? Nope. It must be the Alternate Hypothesis.

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 will compare the same quantities as the Alternate Hypothesis (in this case, μ and ).

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

Do we know we have a normal Distribution?

Yes!! We have a normal distribution because our sample size (50) is big enough---it is at least 30.

Step (*) Since we have a normal distribution, draw a normal curve. Top level: Area Middle level: standard units We always use z-values when we know the population standard deviation, σ. (z)

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

Step (*): Since we have a normal distribution, draw a normal curve. 0 Top level: Area Middle Level: Standard Units (z) Bottom level: Actual Units ($) In this case, the actual units are dollars, since our hypothesis is about the average monthly benefit, which is measured in dollars.

Step (*): Since we have a normal distribution, draw a normal curve. 0 Top level: Area Middle Level: Standard Units (z) Bottom level: Actual Units ($) 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. Standard Units (z) 0 Actual Units ($) Top level: Area.05

Step (*): Once you’ve drawn the picture, start at the Top level and mark off the area in the critical region. Standard Units (z) 0 Actual Units ($).05 Top level: Area

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

We can find the critical value using either Table E or Table F. Click on the table you want to use. Table E gives us the z-values associated with certain areas under the standard normal curve The bottom row of table F gives us the z- values associated with the area in the tail/s.

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. We can subtract the area in the right tail from the total area (1) to get the area to the left! 1-.05=.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 Since they are equally close to.9500, pick the bigger one,.9505.

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

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

Standard Units (z) 0 Actual Units ($) Put critical value here! 1.65

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

Our observed value is That’s bigger than , so it must go here. Why not here? It’s also to the right of , but it’s in the critical region.

How can we figure out whether the observed value will be to the left or right of the critical value?

We can’t compare the observed value and the critical value as long as they are measured in different units. So we’ll convert the observed value to standard units.

The result is called the test value, and we can easily see whether it is bigger or smaller than the critical value!

Standard units (z) 0 Actual units ($) <1.65, so it goes somewhere In the region between 0 and Line up the observed value with the test value; note that it is not in the critical region.

Step 4: Decide whether or not to reject the Null. I throw myself on the mercy of this court.

Standard units (z) 0 Actual units ($) Since the observed value is not in the critical region, we don’t reject the Null.

Step 5: Answer the question. Talk about the claim. Since the claim is the Alternate Hypothesis, use the language of support. We didn’t reject the Null, so we don’t support the claim.

There is not enough evidence to support the claim that benefits increased from 2004 to Let’s see a quick summary! Good idea!

Each click will take you to the next step; step (*) is broken into two clicks. Step 1: Step (*) Standard units (z) 0 Actual units ($) Step Step Step 4: Don’t reject Null. Step 5: There’s not enough evidence to support the claim.

And there was much rejoicing.

Press the escape key to exit the slide show. If you keep clicking through, you’ll see the slides explaining why this was a test about one mean, rather than 2.

It’s true that there are two means in this problem, but there’s a very important difference between them:

We know the mean for 2004, so there’s no need to make and test a hypothesis about it.

It’s true that there are two means in this problem, but there’s a very important difference between them: We know the mean for 2004, so there’s no need to make and test a hypothesis about it. We don’t know the mean for 2005, so we have to form a hypothesis about it and test that hypothesis.

We say the hypothesis test is about one mean when there is just one mean that we don’t know, even if we are comparing it to a known mean. When we say a test is about two means, that will indicate that there are two means and we don’t know either of them. Click anywhere on this slide to return to the hypothesis test. Don’t just hit the space bar or you’ll go to the wrong slide!

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 ($) Put critical value here!

Standard Units (z) 0 Actual Units ($) Put critical value here! 1.645

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

Our observed value is That’s bigger than , so it must go here. Why not here? It’s also to the right of , but it’s in the critical region.

How can we figure out whether the observed value will be to the left or right of the critical value?

We can’t compare the observed value and the critical value as long as they are measured in different units. So we’ll convert the observed value to standard units.

The result is called the test value, and we can easily see whether it is bigger or smaller than the critical value!

Standard units (z) 0 Actual units ($) <1.645, so it goes somewhere In the region between 0 and Line up the observed value with the test value; note that it is not in the critical region.

Step 4: Decide whether or not to reject the Null. I throw myself on the mercy of this court.

Standard units (z) 0 Actual units ($) Since the observed value is not in the critical region, we don’t reject the Null.

Step 5: Answer the question. Talk about the claim. Since the claim is the Alternate Hypothesis, use the language of support. We didn’t reject the Null, so we don’t support the claim.

There is not enough evidence to support the claim that benefits increased from 2004 to Let’s see a quick summary! Good idea!

Each click will take you to the next step; step (*) is broken into two clicks. Step 1: Step (*) Standard units (z) 0 Actual units ($) Step Step Step 4: Don’t reject Null. Step 5: There’s not enough evidence to support the claim.

And there was much rejoicing.