P-value method

dependent samples

The experimentA group of friends wants to compare two energy drinks.

They agree to meet on consecutive Saturdays to run a mile. One week, they drink “GoGetUm” Energy Drink prior to the run and record their times. The following week, the same group drinks “YippetyDoo” Energy Drink and again records their times.

The results (in minutes) are recorded in the table below.

Runner A B C D E F

GoGetUm time

6.2 8.0 10.1 12.3 7.5 9.2

YippetyDoo time

6.1 8.8 10.2 12.1 7.5 8.0

The question

Suppose that the time it takes to run a mile is approximately normally distributed and that both runs were on an indoor track to ensure comparable conditions.

Is there enough evidence to support the claim that the two times were different?

Use the P-value method with α=.05.

Option to work alone and check answerIf you want to try this problem on your own and just check your answer, click on the math professor to the right.

Otherwise, click away from the professor (or just hit the space bar) and we’ll work through this together.

Set-up

With dependent samples, we subtract first, and then calculate the mean difference.

This is very different from what we do with independent samples. Then, we calculate the means first and then subtract to calculate the difference between the two means.

Calculating the differencesRunner A B C D E F

GoGetUm time

6.2 8.0 10.1 12.3 7.5 9.2

YippetyDoo time

6.1 8.8 10.2 12.1 7.5 8.0

Difference: GoGetUm - YippetyDoo

Calculate the difference in the two times.

We can subtract in either order, but we must be consistent once we’ve chosen the order of subtraction. As shown above, I’ve chosen to subtract the “YippetyDoo” time from the “GoGetUm” time.

.1 − .8 − .1 .2 0 1.2

What data to enter into your calculatorRunner A B C D E F

GoGetUm time

6.2 8.0 10.1 12.3 7.5 9.2

YippetyDoo time

6.1 8.8 10.2 12.1 7.5 8.0

Difference: GoGetUm - YippetyDoo

.1 -.8 -.1 .2 0 1.2

Enter these values into your calculator.

Mean difference and standard deviation𝐷=.1𝑠𝐷=.6449… . .

Make sure that’s a sample standard deviation!

And don’t forget to keep it stored in your calculator so we can call it up later and don’t have to use a rounded value!

Step 1: State the hypotheses and identify the claim

• We are asked to investigate the claim that the two times are different.

• We need to express this in terms of the population mean difference in time,

• When two things are the same and we subtract them, we get 0. So when two things are different and we subtract them, we don’t get 0.

That tells us what both hypotheses will be!

The hypotheses

𝐻0 : 𝜇𝐷=0 ¿𝐻1: 𝜇𝐷≠0 (𝑐𝑙𝑎𝑖𝑚)

always has an equals sign in it.

Step (*)

Draw the picture and mark off the observed value.

WAIT!!Do we know we have a bell-shaped distribution?

We have an (approximately) normal curveYes! We were told to suppose the time it takes to run a mile is approximately normally distributed.

Drawing the picture: top and middle levelsStep (*): • First, draw the picture

Top level: Area

Middle Level: Standard Units (t)

Since we don’t know σ, we will approximate it with s, but we have to compensate for this by using t-values.

Drawing the picture: marking the middle in standard unitsStep (*): • First, 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.

Drawing the picture: adding the bottom levelStep (*): • First, draw the picture

Top level: Area

Middle Level: Standard Units (t) 0

Bottom level: Actual Units (min)In this case, the actual units are minutes.

Drawing the picture: marking the center in actual units

Step (*): • First, draw the picture

Top level: Area

Middle Level: Standard Units (t) 0

Bottom level: Actual Units (min) 0

The number from goes in the center in actual units.

Reminder to work bottom-upThen remember:

The -value MethodP

is ottom-upb

Drawing the picture: adding the observed value

Step (*): • Then, start at the bottom level and mark off the observed value:

Standard Units (t) 0

Actual Units (min) 0Bottom level .1

.1 > 0 so it goes to the right if center. The observed value is always on the bottom level (actual units).

Drawing the picture: the tails

Standard Units (t) 0

Actual Units (min) 0Bottom level .1

.1 > 0 so it goes to the right if center. The observed value is always on the bottom level (actual units).

The boundary of the right tail is directly above the observed value, .1.

The left tail is drawn so that the picture is symmetric.

Step 2: Move up to the middle level and mark off the boundary of the right tail in standard units. To do this, convert the observed value to standard units. The result is called the test value.

Standard Units (t) 0

Actual Units (min) 0 .1

Middle level

Put the test value here!

Calculating the test value𝑡=

𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑒𝑟𝑟𝑜𝑟

¿𝐷−𝜇𝐷

( 𝑠𝐷√𝑛 )¿.1−0

( 𝑠𝐷√6 ) Remember to call up the value that is stored in your calculator so that we don’t round until after we’ve calculated t.

Calculating the test value, slide 2𝑡=

𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑒𝑟𝑟𝑜𝑟

¿𝐷−𝜇𝐷

( 𝑠𝐷√𝑛 )¿.1−0

( 𝑠𝐷√6 )¿ .3797…≈ .380

Adding the test value to the pictureLet’s add it to the picture!!

Standard Units (t) 0

Actual Units (min) 0 .1

Middle level

Put the test value here!

.380

Step 3: Move up to the top level and calculate the area in the two tails. This will be the P-value.

Standard Units (t) 0

Actual Units (min) 0 .1

.380

Top Level: Area

Table FSince our standard units are t-values, we’ll need Table F.

To know which row to look in, we need to calculate the degrees of freedom (d.f.):

d.f. = n-1 = 6-1 = 5

Table F, row for d.f. = 5

Look in the row for d.f. = 5.

Find where the t-value .380 would go.

Finding where t = .380 would go

.380 is smaller than all the t-values in this row, so it would be to the left of all of them. (Note that the t-values increase as we move left to right, so the smallest value will be furthest left.)

Finding the areas that correspond to t-values

• Follow this arrow up to the top of the table to see what this tells us about area.

• Since this is a two-tailed test, look in the row titled “Two tails, α.”

Describing P with an inequality

Remember, the α values in the table are the areas that correspond to the t-values below them. Since our t-value is to the left of all the t-values in the table, our P-value, which is area corresponding to that t-value, will also be to the left of all the areas in the table.

Notice that area DECREASES as we move to the right, so that the area furthest left will be the BIGGEST area. Thus, P > .2

We know enough about P to make the decision

At this point, we don’t know exactly what P is, but we have enough information about it to proceed to the next step.

Step 4: Decide whether or not to reject

Compare P to α.

P α

Comparing P to αP > .2

Since α = .05, which is less than .2, we can add α to this inequality.

¿ .05⏟𝛼

Pα So P > α

Making the decisionGeometrically, P is an area. But it also represents the probability that we would get .1 (or something more extreme) if is true.

And P is “BIG.” (Well, at least it’s bigger than α.)

So we shouldn’t reject---we only reject it when our results are very unlikely, i.e. when P is small.

The decision

Don’t reject

Having considered these arguments…

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

language of “support.”• We didn’t reject , so we don’t

support the claim.

There is not enough evidence to support the claim that the two energy drinks produced different results.

Request for a summary

Could we see all that one more time?

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

Step 1. 𝐻0 : 𝜇𝐷=0 ¿𝐻1: 𝜇𝐷≠0 (𝑐𝑙𝑎𝑖𝑚)

Step (*)

standard units (t)

actual units (min)

0

0 .1

Step 2.380

Step 3

P > .2

Step 4: Don’t reject

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

And there was much rejoicing.