P-value Method

2 means, sigmas unknown

The scenarioSodium levels are measured in millimoles per liter (mmol/L) and a score between 136 and 145 is considered normal.

A student wants to test the claim that the average sodium level is different for men and women.

A sample of 9 men had an average sodium level of 143.3, with a standard deviation of 5.3.

A sample of 10 women had an average sodium level of 139.3 with a standard deviation of 6.1

Source: selected entries from the Data Bank in Bluman, Elementary Statistics, eighth edition

The questionSuppose that sodium levels are normally distributed in both men and women and that the standard deviations for the two populations are different.

Evaluate the claim using the P-value method with α = .05.

Option to work alone and check your answerIf you want to work through this on your own, and just check your answer, click on the student to the left.

Otherwise, click away from him and we’ll work through this together.

Set-upThis is a test about 2 means, the mean sodium level for men and the mean sodium level for women.

Population Sample=? 143.3

Population Sample=? 139.3

The hypotheses will be about these population means.

Step 1:State the hypotheses and identify the claim.

The claim is that:

The average sodium level for men is different from the average sodium level for women.

𝜇𝑚 ≠ 𝜇𝑤

The Alternate Hypothesis

I don’t see an equals sign. This must be the AlternateHypothesis.

The Null HypothesisStep 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 that show up in the Alternate.

𝜇𝑚=𝜇𝑤

Equality Now!

Decision to rewrite hypotheses using subtraction𝐻0 : 𝜇𝑚=𝜇𝑤

𝐻 1: 𝜇𝑚≠𝜇𝑤 (𝑐𝑙𝑎𝑖𝑚)

If we subtract, we’ll be able to see what number will be at the center of our distribution!

Rewriting the hypotheses𝐻0 : 𝜇𝑚=𝜇𝑤

𝐻 1: 𝜇𝑚≠𝜇𝑤

It says here if we subtract “bigger minus smaller” we can ensure we get to work with positive numbers later on. Since the sample mean for men is bigger, let’s subtract “men minus women.”

𝜇𝑚−𝜇𝑤=0𝜇𝑚−𝜇𝑤=0

Step (*)

Draw the picture and mark off the observed value.

Need to check for normalityStep (*):

Draw the picture and mark off the observed value.

Stop!Remember: Never draw the picture without first verifying that you have a normal distribution.

We have normality!In this case, we can keep going because we were told to suppose that sodium levels are normally distributed in both men and women.

Drawing the picture: top and middle levels

Step (*) First, draw the picture.

Top level: Area

Middle level: Standard Units (t) 0

Since we don’t know the population standard deviations, we will approximate them with the sample standard deviations. Whenever we do this, we compensate by using t-values instead of z-values.

Click this person if you want further explanation about t-values.

Drawing the picture: the center in standard units

Step (*) First, draw the picture.

Top level: Area

Middle level: Standard Units (t) 0

0 is always at the center in standard units.

Drawing the picture: the bottom level

Step (*) First, draw the picture.

Top level: Area

Middle level: Standard Units (t) 0

Bottom level: Actual units (mmol/L)

We were told at the beginning that millimoles per liter are the units for sodium levels.

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 (mmol/L) 0

The number from the Null always goes in the center here.

Reminder to work bottom-up

Then remember:

The -value MethodP

is ottom-upb

Calculating the observed difference

Step (*)continued

Once you’ve drawn the picture, start at the bottom level and mark off the observed difference.

Standard Units (t) 0

Actual units (mmol/L) 0Bottom level

Observed difference =

Always subtract sample values in the same order as you subtracted population values in the hypotheses---in this case, “men minus women.”

= 143.3 – 139.3 = 4

Adding the observed difference to the picture

Step (*)continued

Once you’ve drawn the picture, start at the bottom level and mark off the observed difference.

Standard Units (t) 0

Actual units (mmol/L) 0Bottom level 4

4 > 0 so it goes on the right side

Drawing the picture: shading the tails

Step (*)continued

Standard Units (t) 0

Actual units (mmol/L) 0Bottom level 4

4 > 0 so it goes on the right side

Mark off the tail that has 4 as its boundary---this is the right tail. Since this is a two-tailed test, remember to mark off the left tail as well .

Step 2: Move up to the middle level. Convert the observed value to standard units and mark this off.

Standard Units (t) 0

Actual units (mmol/L) 0 4

Middle Level

The observed value converted to standard units is called the test value. It goes here.

Calculating the test value𝑡=𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒

𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑒𝑟𝑟𝑜𝑟

¿ 4⏞143.3− 139.3

−0

√ (5.3)9

2

+(6.1)10

2

hypothesized difference in means¿1.5292…≈1.529

Testvalue

Adding the test value to the pictureFinishing up Step 2….

Standard Units (t) 0

Actual units (mmol/L) 0 4

Middle Level

The test value goes here!

Adding the test value to the picture, slide 2

Finishing up Step 2….

Standard Units (t) 0

Actual units (mmol/L) 0 4

Middle Level

The test value goes here!

1.529

Step 3: Move up to the top level and find the area in the two tails; this is the P-value.

Standard Units (t) 0

Actual units (mmol/L) 0 4

1.529

Top Level (area)

P = total area in both tails.

Table FSince our standard units are t-values, we will locate the area on Table F.

We need to look for our test value, 1.529 in the row that corresponds to the correct degrees of freedom.

Formula for degrees of freedomThe degrees of freedom will be the smaller of:

and

Calculating the degrees of freedom𝑛𝑚−1=9−1=8

𝑛𝑤−1=10−1=9

Calculating the degrees of freedom, slide 2𝑛𝑚−1=9−1=8

𝑛𝑤−1=10−1=9The degrees of freedom is the smaller of 8 and 9; the monkey is excited to tell us that that’s 8.

Looking for the test value on Table F

Look for 1.529 in the row for d.f. = 8.

Since 1.529 is between 1.397 and 1.860, it will fall between these two columns.

Finding a range of areas that correspond to the region where our test value falls

Moving up to the top of the chart, we look in the row for a two-tailed test. The α-values here are the areas of the tails with the corresponding t-values as their cut-offs.

We can see that our t-value puts us in between an area of .2 and .1.

A range in which P lies

.1 < P < .2

I don’t have to know exactly what P is; this should be enough information for me to make my decision.

Step 4: Decide whether or not to reject the Null.

In the picture to the left, an unknown artist captures the Null Hypothesis begging for mercy.

Guidelines for making the decision• Compare P to α.• P = area in the two tails = probability of getting 4 (or a more extreme difference) if the Null Hypothesis is true• α= maximum allowable probability of

making a Type I error (rejecting if it is true)

Comparing P to α.1 < P < .2

α = .05

Where does this fit in?

.05 <

Since α = .05 is less than .1, it is less than P.

The decisionP > α

This means the probability we would get the result we did (or one further from 0) is greater than α. So it’s not so unlikely that we should reject it.

Do not reject the Null.

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

Hypothesis, switch to the language of “support”.

• We did not reject the Null, so we don’t support the claim.There is not enough evidence to support the claim that men and women have different sodium levels.

Request for a summaryI wouldn’t mind seeing all that summarized.

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

Step 1.

Step (*)

0Standard Units (t)

Actual units (mmol/L) 0 4

Step 21.529

Step 3P = area is between .1 and .2

Step 4: Don’t reject the 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 continue to click through

the show, you’ll see an informal explanation of why we use t-values

when we don’t know both σ’s.

Ok, here’s a very informal explanation of t-distributions

• We use a t-distribution when we don’t know σ, the population standard deviation.

• If we did know σ, we’d go ahead and use a normal curve with the usual z-values as standard units.

standard units (z) 0

Explanation of t-values

standard units (z) 0

When we don’t know σ, we have to approximate it with s, the sample standard deviation.

And while approximating σ with s is the best we can do, that doesn’t make it good. In fact, it’s kind of like letting someone stomp all over our lovely normal distribution!

Jumping person smushing the normal curve

SMUSH!

The result is a smushed bell-shaped curve. It turns out, this “smushed normal curve” is our t-distribution.

Comparing the t and z-distributionsThe center is lower, so there’s less area in the middle!

The tails are higher, so there’s more area in the tails!

T-values will be bigger than z-values for the same area

This means we have to go farther from center (more standard units) to get a big area in the middle (for confidence intervals) or a small area in the tails (for hypothesis tests.)

That’s why t-values are always bigger than z-values would be for the same area.

Sample size affects the shape of the distribution

And remember, while approximating always has consequences, big samples lead to better approximations, and thus smaller consequences.

Using a big sample is like letting a small person smush the curve---the curve still changes, but only a little, so it’s much closer to the standard normal curve.

Using a small sample is like letting a really big person smush the curve---it gets really smushed and is very different from the standard normal curve.

Summary of t-distributionsOf course, there’s a rigorous explanation for why the t-distribution works. (Sadly, it doesn’t involve any people jumping on curves and smushing them.)

But the gist of it is this:• Approximating things always has consequences.• The consequence of approximating σ with s is

that we use the t-distribution instead of the standard normal curve.

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