P-value method 2 means, both σ’s known

An economist is comparing credit card debt from two recent years. She has gathered the following data: Year 1 sample mean: $6618 sample size: 35 population standard deviation: $1928 Year 2 sample mean: $9205 sample size: 35 population standard deviation: $1928 Source: data is taken from problem 18, section 9-1 of Bluman, Elementary Statistics, eighth edition

The economist claims that average credit card debt increased from year 1 to year 2. Evaluate her claim using the P-value method with α=.01.

If you want to try this problem on your own, click the kid to the left. Otherwise, click away from the kid, and we’ll work through this together.

Set-up Summarizing the data using mathematical symbols, we get: These are what the hypotheses will be about.

Step 1: State the hypotheses and identify the claim. The claim is that the average credit card debt increased from year 1 to year 2. That is: The debt from year 2 is bigger than the debt from year 1. >

I don’t see an equals sign. That should make this the Alternate Hypothesis, though I suppose it could be the cucumbers on my eyes.

If we subtract, we’ll be able to see what number will be at the center of our distribution. I hope she subtracts “bigger minus smaller” so we get a positive number later, when we work with the sample values!

While the number 0 shows up in the hypotheses, since we subtracted “bigger minus smaller” (year 2 is claimed to be bigger in the Alternate hypothesis, and is actually bigger in the sample data) we have set things up so that we will have a right-tailed test and our observed difference will be positive. If you subtracted in the other order, you’ll be doing a left-tailed test and your observed difference will be negative.

Step (*) Draw the picture and mark off the observed value. Do we know we have a normal distribution?

We do! Both sample sizes are 35, so they are big enough---they are at least 30.

Step (*): First, draw the picture Top level: Area Middle Level: Standard Units (z) We use z-values when we know both population standard deviations.

Step (*): First, draw the picture 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 (*): First, draw the picture Top level: Area Middle Level: Standard Units (z) 0 Bottom level: Actual Units ($) In this case, the actual units are dollars, since our hypotheses are about credit card debt.

Step (*): First, draw the picture Top level: Area Middle Level: Standard Units (z) 0 Bottom level: Actual Units ($) The number from the Null Hypothesis always goes in the center in standard units; that’s because we’re drawing the picture as if the Null is true. 0

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

Step (*): (continued) Standard Units (z)0 Actual units ($)0 Bottom level2587 Mark off the right tail, with its boundary at 2587

Step 2: Move up to the middle level. Convert the observed value to standard units and mark this off. (The value you found is called the test value.) Standard Units (z)0 Actual units ($) Middle level Put your answer here!

Hypothesized difference

Finishing up Step 2: Put the test value at the boundary of the tail in standard units. Standard Units (z)0 Actual units ($)

Step3: Move up to the top level and calculate the area in the tail; this is the P-value. Standard Units (z)0 Actual units ($) Top Level (area) P

We can either use Table E or the calculator to find the P-value. Click on the option you prefer. Table E Calculator Note: the calculator used in this tutorial is the Casio fx-115MS plus.

We can use Table E to find our P-value. If our z-value is on the table, table E will give us the area to the left of it, and we’ll have to subtract that area from 1 to get the area in the tail. But…

5.61 is so BIG, it’s off the chart!

Looks like we’re supposed to use

Standard Units (z)0 Actual units ($) P Label.9999 as the area to the left of the observed value P = =.00001

Please be merciful!

Pα <

Step 5: Answer the question. There is enough evidence to support the claim that credit card debt increased from year 1 to year 2.

Let’s recap!

Each click will give you one step. Step (*) is broken up into two clicks. Step (*) Standard Units (z)0 Actual units ($) Step Step 3 P = Step 5: There’s enough evidence to support the claim.

And there was much rejoicing.

Press the escape key to exit the slide show.

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

Our test is right-tailed, so select the area to the right. RIGHT

You’ll see To enter in the calculated test value after the R(, just hit the “Ans” key and then hit the equals key. You should get 0; there is some area in the right tail, but it is so small that it rounds to 0!

Add the P-value to the picture. Standard Units (z)0 Actual units ($) P P = 0

Please be merciful!

Pα 0.01 <

Step 5: Answer the question. There is enough evidence to support the claim that credit card debt increased from year 1 to year 2.

Let’s recap!

Each click will give you one step. Step (*) is broken up into two clicks. Step (*) Standard Units (z)0 Actual units ($) Step Step 3 P = 0 Step 5: There’s enough evidence to support the claim.

And there was much rejoicing.