Traditional Method

2 proportions

Statement of problemThe President of a homeowners’ association believes that pink flamingos on lawns are tacky, or as she puts it, “detrimental to the aesthetic integrity of the neighborhood.” For years, she has dreamed of passing a ban on these “offensive” decorations, but she has been blocked by homeowners who are very attached to their pink flamingos.

Personally, I think we’re quite attractive.

Statement of problem, slide 2In an effort to determine when best to put forward her proposed ban, she has begun an annual poll of her neighbors’ feelings on the “offending” birds.

Last year, she surveyed 100 homeowners and found that 40 of them found the birds “aesthetically displeasing.” This year, she surveyed 120 homeowners and found that 42 of them found the birds “aesthetically displeasing.”

I’m offended.

Me too. Let’s do a survey of our own on how many flamingos find her displeasing.

Statement of problem, slide 3Distressed by these findings, she believes she is actually losing ground on this issue. She immediately decides to do a statistical analysis to test the theory that the proportion of homeowners who find pink flamingos “aesthetically displeasing” has decreased. Help her out by using the Traditional method with α =.01 to test the claim that the proportion of homeowners opposed to the birds has decreased.

Option to work out problem independentlyIf you’d like to work through this problem on your own, go ahead. Just click the flamingo below when you’re ready to check your answer.

Otherwise, keep clicking through the slideshow and we’ll work through this problem together.

Set-up

Year 1 (last year)

Year 2 (this year)

Step 1: State the hypotheses and identify the claim.

The claim is that “the proportion of homeowners opposed to the birds has decreased.” If this is true, then the proportion opposed to the birds this year is smaller than the proportion last year. Equivalently, the proportion opposed last year is bigger thanthe proportion opposed this year.

or equivalently

Step 1slide 2 Step 1

State the hypotheses and identify the claim.

or equivalently

No hats on those p’s! The hypotheses are about population proportions, not sample proportions!

Step 1, slide 3𝑝1>𝑝2We can use either inequality, but planning ahead, we might prefer the “greater than.” We will eventually want to subtract everything over to one side. This way, if we subtract everything over to the to left side, we’ll get an inequality involving “greater than 0.” “Greater than 0” means “positive,” and I like positive numbers!

Step 1, slide 4𝑝1>𝑝2

Since I don’t see an equals sign, I gather this is

will have an equals sign in it.

But will also compare the same two quantities.

𝐻1 :

𝐻0 :𝑝1=𝑝2

Step 1, slide5

If we subtract everything over to the left side, we’ll get the number that goes at the center of our distribution!

Step 1, slide 6Here’s what that looks like for

𝐻0 :𝑝1=𝑝2

Step 1, slide 7So, our hypotheses are:

𝐻0 :𝑝1=𝑝2 ⟹𝑝1−𝑝2=0𝐻1 :𝑝1>𝑝2 ⟹𝑝1−𝑝2>0

Step (*)

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

STOP!

Checking for normality

Do we know we have a normal distribution?

It’s proportions! We have to check!

Checking for normality, slide 2The basic rule is that for both samples np and nq have to be at least 5.

And we’re supposed to use the HYPOTHESIZED values of p and q in checking this.

But what are the hypothesized values of p and q?

Checking for normality slide 3𝐻0 :𝑝1−𝑝2=0

The number from the hypotheses is the hypothesized value for the difference in the two proportions, not for the population proportion.

But hypothesizes that the proportion for the two years is the same. That means there’s really one proportion that is valid for both years. Both and are estimates of this one true proportion, but the best estimate we can get is the one we get by pooling our data. We call this

Checking for normality, slide 4

We’ll keep this as a fraction since the decimal repeats and we don’t want to round this early in the problem!

Checking for normality, slide 5

𝑞=1−𝑝=110110

−41110⏟

  

=69110

get a common denominator

Checking for normality, slide 6Now use these values of and with both sample sizes to check for normality:

Step (*) drawing the top and middle 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.

Step (*) marking the center in standard units

Step (*): Since we have a normal distribution, draw a normal curve.

Top level: Area

Middle Level: Standard Units (z) 0

The center is always 0 in standard units. Label this whenever you draw the picture.

Step (*) labeling the bottom levelStep (*): Since we have a normal distribution,

draw a normal curve.

Top level: Area

Middle Level: Standard Units (z) 0

Bottom level: Actual ValuesThere are no units for proportions.

Step (*) marking off the center on the bottom levelStep (*): Since we have a normal distribution,

draw a normal curve.

Top level: Area

Middle Level: Standard Units (z) 0

Bottom level: Actual Values 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: Traditional method is 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 Values 0

Top level: Area

.01

This is a right-tailed test since uses a greater than (>) sign.

Step 2: Move down to the middle level and mark off the critical value; this is the boundary of the tail in standard units.

Standard Units (z) 0

Actual Values 0

.01

Middle level

Put critical value here.

Picking Table E or Table FWe can find the critical value using either Table E or Table F. Click on the table you want to use.

TABLE E

Table E gives us the z-values associated with certain areas under the standard normal curve

Table F

The bottom row of table F gives us the z-values associated with the area in the tail/s.

Ok, we’ll use Table E

TABLE E

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.)

0

.01

?

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.

We can subtract the area in the right tail from the total area (1) to get the area to the left!

1-.01=.99

Table ESo now we look for an area of .99 (with 4 decimal places this will be .9900) in the area part of Table E.

AREA

Preparing to zoom in on Table ESo now we look for an area of .99 (with 4 decimal places this will be .9900) in the area part of Table E.

AREA

We’ll have to zoom in.

Finding the critical value on Table EThe area closest to .9900 is .9901.

Our critical value is z = 2.33.

Adding the critical value to the pictureSo now we can add the critical value to our picture.

Standard Units (z) 0

Actual Values 0

.01

Middle level

Put critical value here.

2.33

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

Standard Units (z) 0

Actual Values 0

.01

2.33

Bottom level

Problem: where does the observed value go?

Standard units (z) 0

Actual values 0

.01

2.33

ARGGGH!

.05 > 0 so it’s to the right of center, but which spot does it go in? Is it far enough right that it falls in the critical region?

Solution: Convert observed value to standard units

In order to see where .05 goes, we’ll convert it to standard units so that we can see whether it should go to the left or right of 2.33.

Why am I always the one who has to come up with the good ideas?

Calculating the test value𝑧=

𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟

¿(�̂�1− �̂�2 )−(𝑝1−𝑝2)

√𝑝𝑞( 1𝑛1 +1𝑛2 )

Standard error for the distribution of the difference in sample proportions

Calculating the test value, continued

𝑧=𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒

𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟

¿(�̂�1− �̂�2 )−(𝑝1−𝑝2)

√𝑝𝑞( 1𝑛1 +1𝑛2 )

¿.05−0

√( 41110 )( 69110 )( 1100 + 1120 )

.4 -.35

hypothesized difference

Calculating the test value, concluded

𝑧=𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒

𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟

¿(�̂�1− �̂�2 )−(𝑝1−𝑝2)

√𝑝𝑞( 1𝑛1 +1𝑛2 )

¿.05−0

√( 41110 )( 69110 )( 1100 + 1120 )

¿ .763…≈ .76

Adding the observed value to the picture

Standard units (z) 0

Actual values 0

.01

2.33

Now we can add the observed value to the picture!

.76

.76 < 2.33 so it goes to the left of the critical value.

.05

Line up the observed value, .05, with .76.

Step 4: Decide whether or not to reject

Standard units (z) 0

Actual values 0

.01

2.33.76

.05

.05 is not in the critical region. Don’t reject

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

of support.• We did not reject , so we do not

support .

Statement of conclusionThere is not enough evidence to support the claim that the proportion of homeowners opposed to pink flamingos has decreased.

Request for a summary

Would you mind summarizing all that?

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

Step 1.

Step (*)

Standard units (z) 0

Actual values 0

.01

Step 22.33

Step 3

.76

.05

Step 4: Don’t reject

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

CelebrationAnd there was much rejoicing.

Press the escape key to exit the slide show. If you keep clicking through, you’ll re-calculate the

critical value using Table F instead of Table E.

Using Table F to find the critical value

Look at the top of Table F to determine which column will contain our critical value.

Since this is a one-tailed test, look for α = .01 in this row.

Using Table F to find the critical value, slide 2

Since our standard units are z-values, be sure to go all the way to the bottom row of Table F, the one labeled “z.”

Using Table F to find the critical value, slide 3

2.326

z = 2.326

Adding the critical value to the picture, version 2---using the critical value from Table F

So now we can add the critical value to our picture.

Standard Units (z) 0

Actual Values 0

.01

Middle level

Put critical value here.

2.326

Step 3: version 2, using the critical value from Table FStep 3: Move down to the bottom level and mark off

the observed value: .

Standard Units (z) 0

Actual Values 0

.01

2.326

Bottom level

Problem: where does the observed value go? (version 2, with critical value from Table F)

Standard units (z) 0

Actual values 0

.01

2.326

ARGGGH!

.05 > 0 so it’s to the right of center, but which spot does it go in? Is it far enough right that it falls in the critical region?

Solution: Convert observed value to standard units (version 2, using critical value from Table F)

In order to see where .05 goes, we’ll convert it to standard units so that we can see whether it should go to the left or right of 2.326.

Why am I always the one who has to come up with the good ideas?

Calculating the test value (version 2, after using Table F to get critical value)

𝑧=𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒

𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟

¿(�̂�1− �̂�2 )−(𝑝1−𝑝2)

√𝑝𝑞( 1𝑛1 +1𝑛2 )

Standard error for the distribution of the difference in sample proportions

Calculating the test value, continued (version 2, after using Table F to get the critical value)

𝑧=𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒

𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟

¿(�̂�1− �̂�2 )−(𝑝1−𝑝2)

√𝑝𝑞( 1𝑛1 +1𝑛2 )

¿.05−0

√( 41110 )( 69110 )( 1100 + 1120 )

.4 -.35

hypothesized difference

Calculating the test value, concluded (version 2, after using Table F to get the critical value)

𝑧=𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒

𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟

¿(�̂�1− �̂�2 )−(𝑝1−𝑝2)

√𝑝𝑞( 1𝑛1 +1𝑛2 )

¿.05−0

√( 41110 )( 69110 )( 1100 + 1120 )

¿ .7637…≈ .76 4

Adding the observed value to the picture (version 2, using the critical value found via Table F)

Standard units (z) 0

Actual values 0

.01

2.33

Now we can add the observed value to the picture!

.764

.764 < 2.326 so it goes to the left of the critical value.

.05

Line up the observed value, .05, with .764.

Step 4: Decide whether or not to reject (version 2, after using Table F)

Step 4: Decide whether or not to reject

Standard units (z) 0

Actual values 0

.01

2.33.764

.05

.05 is not in the critical region. Don’t reject

Step 5: Answer the question. (version 2, after using Table F)Step 5: Answer the question.

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

of support.• We did not reject , so we do not

support .

Statement of conclusion (version 2, after using Table F)There is not enough

evidence to support the claim that the proportion of homeowners opposed to pink flamingos has decreased.

Request for a summary (version 2, after using Table F)

Would you mind summarizing all that?

Summary (version 2, after using Table F)Each click will give you one step. Step (*) is broken into two clicks.

Step 1.

Step (*)

Standard units (z) 0

Actual values 0

.01

Step 22.326

Step 3

.764

.05

Step 4: Don’t reject

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

Celebration (version 2, after using Table F)

And there was much rejoicing.