Traditional Method 2 means, dependent samples

A 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).

Before 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 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 how efficient everyone is.

After 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) A5048 B5455 C49 D5553 E4746

The 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.

If 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.

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.

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:

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

Clerk# Entries per hour (Before) # Entries per hour (After) Change in # of entries per hour: after - before A5048 B5455 C49 D5553 E4746 This is the data we’ll work with, so enter it into your calculator: -2, 1, 0, -2, -1

Now 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.

Now use your calculator’s statistics functions to calculate the mean and standard deviation for this data. 2 important notes here!

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

Ok, 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.

average difference will be negative. 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

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

The Null Hypothesis always has an equals sign. And it always compares the same quantities as the Alternate.

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

Do 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.

Step (*) 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.

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.

Step (*): 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.

Step (*): 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.

Then remember: The raditional Method T is op-down T

Step (*): (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.

Since 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.

Although 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.

Our sample size is n=5

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

The critical value is t = 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

Step 3: Standard Units (t) 0 Actual Units (# entries/hr) Bottom Level Hmmmm. Clearly -.8 goes to the left of 0. But how far to the left? Should be it here or here?

To see how far to the left to put -.8, we have to see where it falls in relation to 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.

Remember to call up the value that is stored in your calculator so that we don’t round until after we’ve calculated t.

Standard units (t) 0 Actual units (# entries/hr) The test value (-1.372) is between and 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.

Standard units (t) 0 Actual units (# entries/hr) 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.

Does that mean we did a good job? No, but it wasn’t bad enough to be highly significant.

Let’s review!

Review of Set-up Clerk# Entries per hour (Before) # Entries per hour (After) Change in # of entries per hour: after - before A B54551 C49 0 D E4746 Store in calculator to recall when needed

Each click will give you one step. Step (*) is broken into two steps. Step 1. Step (*) Standard units (t) Actual units (# entries/hr) Step Step Step 4: Don’t reject Null. Step 5: There is not enough evidence to support the claim.

And there was much rejoicing.