Traditional method

2 means, σ’s unknown

The scenarioScientists 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 dataThe results are:

Group 1 = 40.2 seconds

seconds

Group 2

= 1.9 seconds

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.

Option to work the problem alone and check your answerIf 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

The hypotheses will be about 2 means, the mean completion time for mice eating standard mouse fare, and the mean completion time for mice on the special diet.

Remembering that the hypotheses are always about the population values, we will use the symbols in stating the hypotheses.

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.

𝜇2 < 𝜇1

Identifying the claim as the Alternate𝜇2<𝜇1There’s no equals sign.

That’s

Stating the NullStep 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.

𝜇2=𝜇1

Rewriting the hypotheses using subtraction

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!

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

−

The hypotheses, written using subtraction

The claim is that this is bigger, so start the subtraction with .

𝜇1−𝜇2=0𝜇1−𝜇2>0

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?

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

Drawing the picture: top two levelsStep (*): • 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 (*): marking off the center in standard unitsStep (*): • 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 (*), adding the bottom levelStep (*): • 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 (*):marking off the center in actual units

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.

Reminder to work top-downThen remember:

The raditional MethodT

is op-downT

Step (*): marking off the critical regionStep (*): (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

Explanation of why this is a right-tailed testStep (*):(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

Because involves a greater than sign(>), this is a right-tailed test. If we had subtracted the other way, would involve a less than (<) sign and our test would be left-tailed.

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!

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

Top of Table F

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

What would itbe like to have 2 tails?

Choosing the correct column in Table F

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

The need to calculate degrees of freedom

To see which row we want, we need to calculate the degrees of freedom (d.f.). This will be the smaller of and

Calculating the degrees of freedom

𝑛1−1=25−1=24𝑛2−1=26−1=25

This one is smaller.

Finding the critical value

1.318

The critical value is t = 1.318.

Adding the critical value to the pictureTo finish up step 2, we add this to the picture.

Standard Units (t) 0

Actual Units (seconds) 0

.1

1.318

Step 3:

Standard Units (t) 0

Actual Units (seconds) 0

.1

1.318

Move down to the bottom level.

Bottom level

Mark off the observed value. This will be the difference between the sample means: (Be sure to subtract in the same order we used for the hypotheses!)

How do I figure out whether it goes here or there?

The need to convert the observed value to standard units

To see where the observed difference of .1 goes, we have to convert it to standard units so we can compare it to 1.318. When we convert the observed difference to standard units, we are calculating the test value.

Calculating the test value (converting the observed value to standard units)

t

¿ .1⏞40.2− 40.1

−0

√ (1.8)25

2

+(1.9)26

2

hypothesized difference

¿ .1930…≈ .193

Adding the test value and observed value to the picture

Standard units (t) 0

Actual units (seconds) 0

.1

1.318

.198 < 1.318, so it goes somewhereto the left of 1.318.

.198

.1

Line up the observed difference with the test value; note that it is not in the critical region.

Step 4: Decide whether or not to reject .

The suspense is killing me!

The decision

Standard units (t) 0

Actual units (seconds) 0

.1

1.318 .193

.1

Since .1 is not in the critical region, we do not reject

Step 5: Answer the question.• Talk about the claim.• Since the claim is , switch to the

language of “support.”• We did not reject so we do not

support

The answer in plain EnglishThere 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

SummaryEach click will give you one step. Step (*) is broken up into two clicks.

Step 1.

Step (*)

.1

1.318 Step 2 .193.1 Step 3

Step 4: Do not reject

Step 5: There is not enough evidence to support the claim.

CelebrationAnd there was much rejoicing.