Traditional Method

1 mean, sigma unknown

Gallup Poll ResultsIn a national phone survey conducted in May 2012, adults were asked:

Thinking about social issues, would you say your views on social issues are:

1. Very conservative2. Conservative3. Moderate4. Liberal5. Very liberal

Source: The Gallup Poll, accessed online at http://brain.gallup.com/documents/questionnaire.aspx?STUDY=P1205006

The liberal/conservative scale

1 2 3 4 5very

conservativevery

liberal

The mean response was 3.10.

I’mAverage!

The questionThe mayor of a small town believes that the average town resident would also score him/herself at a 3.10 on this scale. She asks 45 randomly selected residents the same question and finds that their average response is 3.27 with a standard deviation of .52.

Evaluate the mayors claim using the Traditional Method with α = .05.

3.10

Mayor’s claim

3.27

Survey results

Option to work alone and check your answer

If you want to try this problem on your own, click on the scale to the right when you’re ready to see the solution and check your answer. (If you want, you can click on the number that best describes you, but no one will be tabulating your response.)

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

1 2 3 4 5very

conservativevery

liberal

Set-up

Let’s summarize what we know:

Population

The population in question is the residents of the town. We don’t know anything about this population, but the hypotheses will be about μ, the average response of all (adult) town residents.

Samplen = 45

s = .52

Step 1:

State the hypotheses and identify the claim.

That’s the claim!

The claim

Translating this into symbols:

μ = 3.10 (claim)

Average resident

score

The Hypotheses

μ = 3.10 (claim)

The equals sign means this is the Null Hypothesis.

𝐻0 :

(Whenever the claim is that two things are equal, the Alternate is that these things are not equal.)

Step (*)Draw the picture and mark off the area in the critical region.

The need to check for normality

Is itBell-shaped?

Uh-oh! The curve drawers have gone on strike! They refuse to draw any bell-shaped curves until they are assured that we do have an (approximately) bell-shaped distribution!

We have normalityThat’s ok! Our sample size was 45! Since we do have an approximately bell-shaped distribution and can go ahead and draw our picture!

Drawing the picture: Top and middle levels

We start by drawing our picture…

Top level: Area

Middle level: standard units (t) 0

Standard units are t-values because we don’t know σ and have to approximate it with s.

Option to click for an explanation of t-values

We start by drawing our picture…

Top level: Area

Middle level: standard units (t) 0

Standard units are t-values because we don’t know σ and have to approximate it with s.

If figuring out whether to use t-values or z-values makes your head spin, click on the person below for an informal explanation. Otherwise, click away from the person (or just hit the space bar) to keep going.

Drawing the picture: marking the center in standard units

We start by drawing our picture…

Top level: Area

Middle level: standard units (t) 0

Standard units are t-values because we don’t know σ and have to approximate it with s.

0 is always the center in standard units.

Drawing the picture: adding the bottom level

We start by drawing our picture…

Top level: Area

Middle level: standard units (t) 0

Bottom level: actual units (points) 3.10

These are points on the 1-5 scale of possible responses.

The number from the Null always goes here.

Reminder to work top-downThen remember:

The raditional MethodT

is op-downT

Step (*) starts at top level

Now start at the Top level and mark off the area in the critical region.

standard units (t) 0

actual units (points) 3.10

Top level: Area

Marking off the area in the tails

standard units (t) 0

actual units (points) 3.10

Top level: Area

The Alternate Hypothesis is that , so this is a 2-tailed test.

α = .05 = total area in both tails

This means = .025 is the area in each tail.

.025.025

Step 2

standard units (t) 0

actual units (points) 3.10

.025.025

Move down to the middle level and mark off the critical values; these will be the boundaries of the tails in standard units.

Middle level:

Critical values go here

Table F

Since our standard units are t-values we’ll use Table F to get the critical values.

We need to know which row to look in, so we need to calculate the degrees of freedom.

Calculating degrees of freedom

d.f. = n-1 = 45-1 =44 d.f.

So we look for 44 in the column labeled “d.f.”.

It ought to be here, but it isn’t!

Rule for d.f. not on the table

d.f.That’s ok; the rule is when the value we want isn’t on the chart we choose the closest smaller value.

Table F: just the headings and a few rows

Missing Rows

Note: the missing rows were deleted solely to make the part of the Table we need fit better on the slide.

Finding the critical values

Missing Rows

• Look in the row for d.f. = 40.• At the top of the table, look for α = .05 in the row for two-tailed

tests. Then look in the column below this.

2.021

Finding the critical values, slide 2

Missing Rows

2.021

𝑡=±2.021• We get the absolute value from Table F.• The plus/minus is because of the position of

the critical values; in standard units anything to the left of center is negative and anything right of center is positive.

Adding the critical values to the picture:

standard units (t) 0

actual units (points) 3.10

.025.025

Middle level:

Critical values go here

-2.021 2.021

Step 3Move down to the bottom level and mark off the observed value

standard units (t) 0

actual units (points) 3.10

.025.025

2.021-2.021bottomlevel

The difficulty of seeing where the observed value goes

standard units (t) 0

actual units (points) 3.10

.025.025

2.021-2.021bottomlevel

Argh!! I can see that 3.27 points is bigger than 3.10 points, but in order to see whether it falls in the critical region or not I need to know how it compares to 2.021 standard units!

3.27 3.27

Which one is right?

The need to calculate the test value

On order to see whether 3.27 is to the left or right of the critical value, we have to convert it to standard units. The result is called the test value.

Observed value (in points)

conversionformula

Test value(in standard units)

Calculating the test value

𝑡=𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒

𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑒𝑟𝑟𝑜𝑟

¿ 𝑋−𝜇

( 𝑠√𝑛 )

Hypothesized value of μ

2.193

3.27 points

conversionformula

t = 2.193

Now we can add the test value and observed value to the picture!

standard units (t) 0

actual units (points) 3.10

.025.025

2.021-2.021 2.193

2.193 > 2.021 so 2.193 goes to the right of 2.021

3.27

Line up 3.27 with 2.193

Step 4: Decide whether or not to reject the Null

standard units (t) 0

actual units (points) 3.10

.025.025

2.021-2.021 2.193

3.27

The observed value is in the critical region; reject the Null.

Step 5: Answer the question in plain English

blah blahI hate all this technical language!

The decision in plain English

• Remember to talk about the claim.• Since the claim is the Null, stick with the

language of “rejection.”

There is enough evidence to reject the claim that the town’s residents give themselves an average rating of 3.10.

Let’s recap…

Could we see a quick re-cap?

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

Step 1.

Step (*)

standard units (t) 0

actual units (points) 3.10

.025 .025

Step 22.021-2.021

Step 3

2.193

3.27

Step 4: Reject the Null.

Step 5: There’s enough evidence to reject the claim.

And there was much rejoicing.

Press the escape key (“esc”) to exit the slide show. If you keep clicking through, you’ll go to the informal explanation of t-distributions.

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

Curve-smushing monsters!

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 a monster with very big feet stomp all over our lovely normal distribution!

Curve-smushing monsters, slide 2!

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 monster 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 monster 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 mathematical explanation for t-distributions. (Sadly, it doesn’t involve any monsters!)

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.

Slide to take you back to the main problem

Follow me!

Click anywhere on this slide to return to the main problem.

Don’t just hit the space bar or you’ll exit the slide show!