Traditional method 2 means, σ’s unknown

Scientists studying the effect of diet on cognitive ability are comparing two groups of mice. The first group is fed standard mouse fare, and the second group is fed a special diet designed to improve cognitive ability. After a week on their respective diets, the mice are placed in a maze and their completion times are measured.

The results are: Suppose that maze completion times for both populations is normally distributed, and that the population variances for mice on the two diets are unequal. Use the traditional method with α =.1 to evaluate the claim that group 2 (the mice who are fed the special diet) completes the maze in a shorter time.

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

Set-up

Step 1: State the hypotheses and identify the claim. The claim is that group 2 completes the maze faster. This means: The time it takes group 2 is less than the time it takes group 1. <

There’s no equals sign.

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 always compare the same quantities as the Alternate.

If we subtract one of the terms over, we’ll get the number zero showing up in the hypotheses. Yes! And this will be the number that shows up in the center of our distribution!

And, if we subtract “bigger time – smaller time,” we’ll get a positive number later, when we work with the sample values!

We get 0 in the hypotheses, but since we subtracted “bigger - smaller” we’ll get a positive number when we work with the sample data. We could subtract the other way, but it’s nice to plan ahead and make things easier.

Step (*) Draw the picture and label the area in the critical region. Do we know we have a bell-shaped distribution?

We do! While both samples are small (fewer than 30), we are told that maze completion time is normally distributed in both populations.

Step (*): Draw the picture Top level: Area Middle Level: Standard Units (t) We always use t-values when we don’t know both σ’s; in this case, we don’t know either one.

Step (*): Draw the picture Top level: Area Middle Level: Standard Units (t) 0 The center is always 0 in standard units. Label this whenever you draw the picture.

Step (*): Draw the picture Top level: Area Middle Level: Standard Units (t) 0 Bottom level: Actual Units (seconds) In this case, the actual units are seconds, since our hypothesis is about the average difference in maze completion time, measured in seconds.

Step (*): First, draw the picture Top level: Area Middle Level: Standard Units (t) 0 Bottom level: Actual Units (seconds) 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 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 (seconds) 0 Top level: Area.1

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 (points) 0 Top level: Area.1

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 (t) 0 Actual Units (seconds) 0.1 Middle Level Put critical value here!

To find the critical value, we’ll need table F.

Since this is a one-tailed test, look for α in this row. What would it be like to have 2 tails?

α =.1 We’ll find our critical value in this column.

This one is smaller.

1.318 T he critical value is t =

To finish up step 2, we add this to the picture. Standard Units (t) 0 Actual Units (seconds)

Step 3: Standard Units (t) 0 Actual Units (seconds) Move down to the bottom level. Bottom level How do I figure out whether it goes here or there?

To see where the observed difference of.1 goes, we have to convert it to standard units so we can compare it to When we convert the observed difference to standard units, we are calculating the test value.

hypothesized difference

Standard units (t) 0 Actual units (seconds) < 1.318, so it goes somewhere to the left of Line up the observed difference with the test value; note that it is not in the critical region.

The suspense is killing me!

Standard units (t) 0 Actual units (seconds)

Step 5: Answer the question.

There is not enough evidence to support the claim that mice on the special diet complete the maze faster. Let’s review!

Standard units (t) 0 Actual units (seconds) 0 Each click will give you one step. Step (*) is broken up into two clicks. Step (*) Step Step 3 Step 5: There is not enough evidence to support the claim.

And there was much rejoicing.