Traditional Method

2 means, dependent samples

The problemA data entry office finds itself plagued by inefficiency. In an attempt to improve things the office manager calls in a group of consultants to do a workshop in Streamlining and Learning to Organize (SLO for short).

The problem, continuedBefore the consultants arrive, the manager records the productivity of 5 of the office’s data entry clerks. She keeps track of the average number of new entries each clerk can process in one hour.

The consultants

The consultants come in with lots of enthusiasm,

lots of jargon, Blah, blah,

blahblah,

blah….

lots of new processes,

and lots of forms for keeping track of just howefficient everyone is.

The dataAfter the new processes are in place, the manager again records the average number of entries each clerk can process in one hour. The results are summarized in the table below:

Clerk # Entries per hour (before)

# Entries per hour (after)

A 50 48

B 54 55

C 49 49

D 55 53

E 47 46

Final statement of the problemThe office manager is now concerned that the new processes have actually made the office less efficient. Suppose that the number of entries a person can process in an hour is approximately normally distributed and test this concern using the traditional method with α =.01.

Option to work out and check answerIf you want to work this problem out and just check your answer, go ahead. Click on the brain to the right when you’re ready to check your answer.

Otherwise, click away from the picture (avoid the entire image!) or just hit the space bar and we’ll work through it together.

Set-up• There’s a lot to set up in this problem, because we

are working with dependent samples---the “before” group and the “after” group.

• The “before” group and the “after” group are actually the same set of employees, so we can pair up each employee with him/herself and calculate how much that individual employee’s productivity changed.

Reminder to subtract first, then calculate the mean Remember: with independent

samples, we calculate the means first and then subtract to calculate the difference between the two means.

With dependent samples, we subtract first, and then calculate a single mean---in this case the mean change in entries processed per hour.

Choosing the order of subtraction• So our first step is to subtract the “before” and “after” values for each clerk.

• We can subtract in either order, but we must be consistent in the order.

• If we subtract “after” minus “before” then the sign of the result will match the direction of change:

positive result # entries per hour increasednegative result # entries per hour decreased

Choosing the order of subtraction, continued

positive result # entries per hour increasednegative result decreased

That’s nifty! Let’s subtract “after minus before”!

Subtracting “after minus before”Clerk # Entries per

hour (Before)# Entries per hour (After)

Change in # of entries per hour: after - before

A 50 48B 54 55C 49 49D 55 53E 47 46

48−50=−255−54=149−49=053−55=−246−47=−1

This is the data we’ll work with, so enter it into your calculator:

-2, 1, 0, -2, -1

Calculating the mean differenceNow use your calculator’s statistics functions to calculate the mean and standard deviation for this data.

This means that, on average, each employee processed .8 fewer entries per hour.

Calculating the sample standard deviationNow use your calculator’s statistics functions to calculate the mean and standard deviation for this data.

average change in # of entries processed

= standard deviation for data = 1.303….

2 important notes here!

Notes on standard deviationNow use your calculator’s statistics functions to calculate the mean and standard deviation for this data.

average change in # of entries processed

= standard deviation for data = 1.303….

2 important notes here!

1. Be sure to calculate this as a sample standard deviation.

2. Don’t round! We will use this number in future calculations, and we never want to round until the very last step. So keep it stored in your calculator so you can recall it when you need it.

Transition from set-up to hypothesis testOk, now that we have our data set up, let’s move on to the familiar six steps of hypothesis testing.

Step 1:State the hypotheses and identify the claim.• We are asked to evaluate the “concern” that the

office is now less efficient. • A less efficient office will process fewer entries

per hour.• This will be the case if the average number of

entries processed decreases , after the consultants come in, which means that the average difference will be negative.

𝜇𝐷

The claimaverage difference will be negative.

𝜇𝐷

Be sure to use the symbol for population mean difference. The hypotheses are always about the population, never the sample. After all, we know the sample mean is negative; we are trying to see if this is evidence that the consultants were counter-productive overall.

< 0

The Claim is the Alternate Hypothesis𝜇𝐷<0

I can’t see an equals sign. I think we’ve located the Alternate Hypothesis!

The Null Hypothesis

The Null Hypothesisalways has an equals sign.

𝐻0 :=¿

And it always compares the same quantities as the Alternate.

𝜇𝐷0

Step (*)

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

Wait!!

Need to check for normalityDo we know we have a normal distribution?

Concluding that we do have a normal distributionDo we know we have a normal distribution?

Remember, we supposed that the number of entries a clerk can process in an hour is approximately normally distributed.

WE DO

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

draw a normal curve.

Top level: Area

Middle level: standard units (t)

We always use t-values when we don’t know the population standard deviation, σ.

Drawing the picture: labeling the center in standard units

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

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 (*): Since we have a normal distribution,

draw a normal curve.

Top level: Area

Middle Level: Standard Units (t) 0

Bottom level: Actual Units (# entries/hr)

In this case, the actual units are # of entries per hour.

Drawing the picture: labeling the center in actual unitsStep (*): Since we have a normal distribution, draw a normal curve.

Top level: Area

Middle Level: Standard Units (t) 0

Bottom level: Actual Units (# entries/hr) 0

The number from the Null Hypothesis goes at the center in actual units.

Reminder to work top-downThen remember:

The raditional MethodT

is op-downT

Drawing the picture: marking 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.

Standard Units (t) 0

Actual Units (# entries/hr) 0

Top level: Area .01

This is a left-tailed test because the Alternate Hypothesis involves a less than sign (<). α = .01 = area in tail

Step 2:

Standard Units (t) 0

Actual Units (# entries/hr) 0

.01

Middle Level

Critical value goes here!

Move down to the middle level and mark of the critical value, which is the boundary of the left tail.

Table FSince our standard units are t-values, we find the critical value using Table F.

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

Discussion of sample sizeAlthough in a sense we have two samples (“before” and “after”), we have only one sample size.

With dependent samples, both groups must be the same size because we pair off the individuals in the two groups in order to calculate the difference between them.

In the case of “before and after,” like we have here, the two groups are actually exactly the same people, just measured at different times.

BEFORE AFTER

Sample sizeOur sample size is n=5

Degrees of freedomThe degrees of freedom is one less than this:

Using Table F

So we look the in the row for d.f. = 4.

Since this is a one-tailed test with α = .01, look in this column.

3.747

The critical valueThe critical value is t = -3.747.

Don’t forget the negative sign! Remember, Table F gives us the absolute value of t; any t-value to the left of center will be negative.

Finishing up Step 2:

Standard Units (t) 0

Actual Units (# entries/hr) 0

.01

Critical value goes here!

Middle Level -3.747

Step 3: Move down to the bottom level and marked off the observed value. This is the mean sample difference,

Standard Units (t) 0

Actual Units (# entries/hr) 0

.01

-3.747Bottom Level

Hmmmm. Clearly -.8 goes to the left of 0. But how far to the left? Should be it here or here?

The test value

To see how far to the left to put -.8, we have to see where it falls in relation to -3.474. But it’s difficult to compare these values as long as they are in different units. So we will convert -.8 to standard units to make the comparison easier.

The result of converting the observed value to standard units is called the test value.

Calculating the test value, slide 1𝑡=

𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑒𝑟𝑟𝑜𝑟

¿𝐷−𝜇𝐷

( 𝑠𝐷√𝑛 )¿− .8−0

( 𝑠𝐷√5 ) 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 )¿− .8−0

( 𝑠𝐷√5 )¿−1.3719…≈−1.372

Adding the test value and observed value to the picture

Standard units (t) 0

Actual units (# entries/hr) 0

.01

-3.747 -1.372

The test value (-1.372) is between -3.747 and 0.

-.8

Line up -.8 with the test value. We can see that it does not fall in the critical region.

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

The decision

Standard units (t) 0

Actual units (# entries/hr) 0

.01

-3.747 -1.372

-.8

Since -.8 is not in the critical region, don’t reject the Null.

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

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

not support the Alternate.

There is not enough evidence to support the claim that the consultants made the company less efficient.

Discussion of decision

Does that mean we did agood job?

No, but it wasn’t bad enough to be highly significant.

Let’s review!

Review of Set-up:Review of Set-upClerk # Entries per

hour (Before)# Entries per hour (After)

Change in # of entries per hour: after - before

A 50 48 -2

B 54 55 1

C 49 49 0

D 55 53 -2

E 47 46 -1

Store in calculator to recall when needed

Summary of hypothesis testEach click will give you one step. Step (*) is broken into two steps.

Step 1.

Step (*)

Standard units (t)

Actual units (# entries/hr)

0

0

.01

Step 2-3.747

Step 3

-1.372

-.8

Step 4: Don’t reject Null.

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

CelebrationAnd there was much rejoicing.