traditional method one proportion. a researcher claims that the majority of the population supports...

Traditional Method

One Proportion

The problemA researcher claims that the majority of the population supports a proposition raising taxes to help fund education.

In a survey of 100 people, 52 said they supported the proposition.

Use the Traditional method with α=.05 to evaluate the researcher’s claim.

Option to try the problem on your ownIf you want to try this problem on your own and just check your results, click on Leonardo to the right. Otherwise, click away from him and we’ll work through this together.

Set-upThis is a test about 1 proportion, the proportion of voters who support the proposition.

Here’s what we know:Population

p= ?

This is what the hypotheses will be about!

Set-up continued: sample valuesSet-upThis is a test about 1 proportion, the proportion of voters who support the proposition.

Here’s what we know:Population

p= ?

Step 1:State the hypotheses and identify the claim.

We are asked to evaluate the claim that the majority of the population supports the proposition. That is:

𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 h𝑤 𝑜𝑠𝑢𝑝𝑝𝑜𝑟𝑡> .5That’s

p!

Determining that the claim is the Alternate

p > .5

Step 1, continued

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

Step 1 concluded: both hypothesesStep 1

• The Null Hypothesis has to have an equals sign, since the Null always claims there is no difference between things.

• The Null Hypothesis has to use the same number that shows up in the Alternate.

𝐻0 : 𝑝=.5

Step (*)

Draw the picture and label the area in the critical region.

Whoa!

The need to check for normalityDo we know we have a normal distribution?

With proportions, we have to check!

Checking for normalityRemember: n = 100

p = .5

Be sure to use the hypothesized value of p, not q = 1 - p

= 1 - .5 =.5

To check for normality:

np = (100)(.5) = 50

Drawing the picture: top two levelsStep (*) Since we have a normal distribution,

draw a normal curve.

Top level: Area

Middle level: standard units (z)

We always use z-values when we are working with proportions.

Drawing the picture; the center in standard unitsStep (*): • Since we have a normal

distribution, draw the picture

Top level: Area

Middle Level: Standard Units (z) 0

The center is always 0 in standard units.

Drawing the picture: bottom levelStep (*): • Since we have a normal distribution, draw the picture

Top level: Area

Middle Level: Standard Units (z) 0

Bottom Level: Actual valuesThere are no units for proportions.

Drawing the picture: the center of the bottom level

Step (*): • Since we have a normal distribution, draw the picture

Top level: Area

Middle Level: Standard Units (z) 0

Bottom Level: Actual values .5

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: Traditional is top-downThen remember:

The raditional MethodT

is op-downT

Step (*): marking off the area in the critical regionStep (*):continued

Once you’ve drawn the picture, start at the Top level and label the area in the critical region.

Top level: Area

Middle Level: Standard Units (z) 0

Bottom level: Actual Values .5

Step (*): the area in the right tailStep (*):continued

Once you’ve drawn the picture, start at the Top level and label the area in the critical region.

Top level: Area

Middle Level: Standard Units (z) 0

Bottom level: Actual Values

.05

This is a right-tailed test since includes a greater than (>) α=.05 = area in right tail

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 (z) 0

Actual Values .5

.05

Middle Level

Put critical value here!

Choosing between table E and table F.To find the critical value, we can use either

Table Eor

Table FClick on the name of the table you would like to use.

Setting up to use Table EOur picture looks like this: (we know the area to the right of the critical value, and want to know the critical value.)

.05

0?

To use Table E, we want to have our picture match this one, where we know the area to the left of the critical value.

Subtract the area in the right tail from the total area (1) to get the area to the left.

1-.05=.95

Table E

Now we can look up .9500 in the area part of Table E.

Area

Let’s zoom in!

Table E: finding areas closest to .95The two areas closest to .9500 are .9495 and .9505. They are exactly the same distance from .9500; when this happens, we pick the bigger area, .9505.

Z-value associated with .9505

The z-value associated with the area .9505 is 1.65.

Finishing up step 2:

Standard Units (z) 0

.05

Put critical value here!

Actual Values .5

Finishing up step 2: Add the critical value to the picture.(Table E version)

Finishing up step 2:Add the critical value to the picture.

Standard Units (z) 0

.05

Put critical value here!

Actual Values .5

1.65

Step 3, Table E VersionStep 3:

Move down to the bottom level.Mark off the observed value (.

Standard Units (z) 0

Actual Values .5

.05

1.65

Bottom level

Question: where does the observed value go? (Table E version)

Step 3: Move down to the bottom level.Mark off the observed value (.

Standard Units (z) 0

Actual Values .5

.05

1.65

Bottom level

Our observed value is .52. That’s clearly bigger than .5, but should it go here or here?

The need to convert the observed value to standard units (Table E version)

We want to compare our observed value, .52, to our critical value, 1.65. But 1.65 is in standard units, and .52 isn’t.

So we convert .52 to standard units in order to make the comparison possible.

Converting the observed value to standard units (Table E version)

This is the formula for the standard error in the distribution of sample proportions.

Converting the observed value to standard units, continued (Table E

version)

¿.52− .5

√ ( .5 )(.5)100

= .4

Plotting the observed value (Table E version)

Standard units (z) 0

Actual values .5

.05

1.65

.4 < 1.65, so falls to the left of 1.65

.4

Line up the observed value with the test value; note that it is in the critical region.

.52

Step 4:Decide whether or not to reject . (Table E version)Step 4:

Decide whether or not to reject .

The decision (Table E version)

Standard units (z) 0

Actual value 5

.05

1.65.4

.52

Since the sample proportion is not in the critical region, don’t reject

The Null expresses joy (Table E version)

𝐻0

Hooray!I hate being rejected!

Most of my patients do.

Step 5: Answer the question. (Table E version)

Step 5: Answer the question.

• Talk about the claim.• Since the claim is , use the language

of support.• We did not reject , so we don’t

support .There is not enough evidence to support the claim that a majority of the population supports the proposition.

Let’s see it again (Table E version)Let’s see this one more time!

Summary (Table E version)Each click will take you to the next step; step (*) is broken into two clicks.

Step 1: 𝐻0 :𝑝= .5𝐻1 :𝑝> .5(𝑐𝑙𝑎𝑖𝑚)

Step (*)

Standard units (z) 0

Actual values .5

.05

Step 21.65

Step 3

.4

.52

Step 4: Don’t reject .

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

Final slide of Table E version

And there was much rejoicing!

That’s all!

Press the escape key to exit the slide show. Don’t hit the mouse or press the space bar.

Using Table F to find the critical value:

We need to look at the top part of Table F to determine which column will contain our z-value.

Since this is a one-tailed test, look for α = .05 in this row.

Using Table F to find the critical value, slide 2

Be sure to go all the way to the bottom row of Table F; this is the only row that gives us z-values!

z = 1.645

Finishing up step 2:

Standard Units (z) 0

Actual Units .5

.05

Put critical value here!

Finishing up step 2: Add the critical value to the picture.

Standard Units (z) 0

.05

Put critical value here!

Actual Values .5

1.645

Step 3: Move down to the bottom level.Mark off the observed value (.

Standard Units (z) 0

Actual Values .5

.05

1.645

Bottom level

Question: where does the observed value go?Step 3: Move down to the bottom level.Mark off the observed value (.

Standard Units (z) 0

Actual Values .5

.05

1.645

Bottom level

Our observed value is .52. That’s clearly bigger than .5, but should it go here or here?

The need to convert the observed value to standard unitsWe want to compare our observed value, .52, to our

critical value, 1.645. But 1.645 is in standard units, and .52 isn’t.

So we convert .52 to standard units in order to make the comparison possible.

Converting the observed value to standard units

This is the formula for the standard error in the distribution of sample proportions.

Converting the observed value to standard units, continued

¿.52− .5

√ ( .5 )(.5)100

= .4

Plotting the observed value

Standard units (z) 0

Actual values .5

.05

1.645

.4 < 1.645, so falls to the left of 1.645

.4

Line up the observed value with the test value; note that it is in the critical region.

.52

Step 4:Decide whether or not to reject .

The decision

Standard units (z) 0

Actual value 5

.05

1.645.4

.52

Since the sample proportion is not in the critical region, don’t reject

The Null expresses joy

𝐻0

Hooray!I hate being rejected!

Most of my patients do.

Step 5: Answer the question.• Talk about the claim.• Since the claim is , use the language

of support.• We did not reject , so we don’t

support .There is not enough evidence to support the claim that a majority of the population supports the proposition.

Let’s see it againLet’s see this one more time!

SummaryEach click will take you to the next step; step (*) is broken into two clicks.

Step 1: 𝐻0 :𝑝= .5𝐻1 :𝑝> .5(𝑐𝑙𝑎𝑖𝑚)

Step (*)

Standard units (z) 0

Actual values .5

.05

Step 21.645

Step 3

.4

.52

Step 4: Don’t reject .

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

Final slide

And there was much rejoicing!