Welcome to the Unit 5 Seminar for MM305!

I hope you have had a good evening so far.

I have a lot of information to share with you tonight. We maynot be able to go over all of them in our seminar but I havewritten them (hopefully) straightforward enough that you cango over them after our seminar and follow the concept. Mynotes will give you the big picture and of course for a littlemore detail you also need to read the book along with it.

It is not too late to catch up in this class but you need to hurryup a little as concepts are going to get a little busier and youneed to spend more time to understand them. It won’tnecessarily get harder, just busier. I don’t mean to scare you.I just want you to be aware of it so you can plan your timeaccordingly.

Need Help?

Please use all the options to get help in this class. You can useour office hours on Mondays and Wednesdays (8:00 pm to9:00 pm ET on AIM) to get a one-on-one help.

You can also email me your questions but I really prefer thatyou post your questions on the board (under Any Questionslink) so other students can also benefit from the questions andtheir answers.

You can also use the NetTutor online tutoring service that issponsored by Kaplan. To access their service just click on theNetTutor icon on your "MyDesk" page.

Anyone is using them on a regular basis and if so, are youhappy with the service? I know I have talked about this beforebut it won’t hurt to share it again.

Excel Note:

It seems most of you are using excel and are not experiencingtoo many problems. The problems will become a little morechallenging and excel will really help. Do practice though –itwill help.

You can get information about any statistical procedure bytyping the name of the procedure in the HELP command ofExcel. You will get an explanation and example for thatcommand or procedure (i.e. mean, standard deviation,regression) .

Definition: Point Estimates

In most cases we don’t know the mean and standard deviation of a particular parameter of interest (like height) of a large population(think of census data, for example). So, we get an estimate of thosevalues by getting a sample from the population and calculating themean and standard deviation of that sample.

We call these sample mean and sample standard deviation valuesthe “Point Estimates” of the population mean and populationstandard deviation. (A point estimate is just a single number used to estimate some parameter of a population.

Definition: Interval Estimates

Since these sample means and standard deviations may notbe very accurate (i.e., the sample may not reflect the goodsample from the population) then we want to set an intervalaround the value of sample mean and express that thisinterval contains true population mean with a certain degreeof accuracy. This is called confidence interval.

μ

Sampling from a population

Suppose we draw 20 samples from a population and calculatethe mean of each. We would expect only 1 in 20 to be outsideof the interval that is 2 standard deviations above and below the mean. (The arrow is pointing to the sample where itsmean is outside the confidence interval.)

μ

95% Confidence Interval

Now, suppose we draw a single sample from the population. Also suppose we then build a 2 standard deviation intervalaround the value obtained. In most cases, the true mean ofthe population will be within the interval. (The exceptionwould be the sample that falls outside of the interval aroundthe true mean.

μ

Example: Confidence Interval

Suppose we observe that, in a sample of 50 commuters, theaverage length of travel to work is 30 minutes with apopulation standard deviation of 2.5 minutes.

What is the standard error for the sampling distribution?

Answer: 2.5 / sqrt(50) = 0.35

Example: Confidence Interval

Suppose we observe that, in a sample of 50 commuters, theaverage length of travel to work is 30 minutes with apopulation standard deviation of 2.5 minutes.

What is the standard error for the sampling distribution?

Answer: 2.5 / sqrt(50) = 0.35

To create the 95% confidence interval, you would take 2standard errors and subtract and add it to the mean.

[ 30 – 0.7, 30 + 0.7] = [29.3, 30.7]

We can be 95% confident that the true population mean iswithin that interval.

Confidence Interval

Suppose we observe that, in a sample of 100 cereal boxes,the average weight of the cereal is 26.8 ounces with aPopulation 2 ounces.

Everyone: What is the standard error for the samplingdistribution?

Confidence Interval

Suppose we observe that, in a sample of 100 cereal boxes,the average weight of the cereal is 26.8 ounces with aPopulation 2 ounces.

Everyone: What is the standard error for the samplingdistribution?

Answer: 2 / √(100) = 0.2

Confidence Interval

Suppose we observe that, in a sample of 100 cereal boxes,the average weight of the cereal is 26.8 ounces with aPopulation 2 ounces.

Everyone: What is the standard error for the samplingdistribution?

Answer: 2 / √(100) = 0.2

Everyone: Construct a 95% confidence interval for the mean.

Confidence Interval

Suppose we observe that, in a sample of 100 cereal boxes,the average weight of the cereal is 26.8 ounces with aPopulation 2 ounces.

Everyone: What is the standard error for the samplingdistribution?

Answer: 2 / √(100) = 0.2

Everyone: Construct a 95% confidence interval for the mean

Answer: 2 standard errors = 0.4 so subtract it from and add itto the mean.

[ 26.8 – 0.4, 26.8 + 0.4] = [26.4, 27.2]

We can be 95% confident that the true population mean iswithin that interval.

Excel: Confidence Interval

Suppose we observe that, in a sample of 50 commuters, theaverage length of travel to work is 30 minutes with apopulation standard deviation of 2.5 minutes.

Click on a cell and then type =CONFIDENCE(0.05,2.5,50) inthe Excel input box and click on ok. We only use the samplesize and standard deviation in this command. You will get thevalue of 0.692951.

So, the expression =CONFIDENCE(0.05,2.5,50) equals0.692951 or rounded to 0.7. Therefore, the interval of theaverage length of travel to work (30 minutes) is calculated as: 30 +/- 0.7 minutes. This results in an interval of 30 + 0.7= 30.7 and 30 – 0.7 = 29.3.

We are 95% confident that the commute time interval is from 29.3 to 30.7 minutes

30 - 0.7 = 29.3 and 30 + 0.7 = 30.7

Everyone: Use Excel to determine a Confidence Interval

Suppose we observe that, in a sample of 100 cereal boxes,the average weight of the cereal is 26.8 ounces with aPopulation standard deviation of 2 ounces. Use excel toconstruct a 95% Confidence Interval for the mean.

Everyone: Use Excel to determine a Confidence Interval

Suppose we observe that, in a sample of 100 cereal boxes,the average weight of the cereal is 26.8 ounces with aPopulation standard deviation of 2 ounces. Use excel toconstruct a 95% Confidence Interval for the mean.

Answer:=CONFIDENCE(0.05,2,100)

Using Excel, the above is equal to 0.391993. The confidence interval is therefore:

[ 26.8 – 0.39, 26.8 + 0.39 ] = [ 26.41, 27.19 ]

Hypothesis Testing

Basically, t test statistic and Z test statistic are used inHypothesis testing to reject or accept a claim. The claim isusually Null Hypothesis (called H0) and if we reject H0 weautomatically accept Alternative Hypothesis (called H1)because that is the only other option (kind of like plan B)available to us.

Null and Alternative hypothesis are kind of complement ofeach other. For example, if Null hypothesis claims that meanvalue of something is less than or equal to a certain value(book call this directional) then alternative would be meanvalue is greater than that value. Or, if Null says mean isequal to a certain value then Alternative says mean is NOTequal to that value. Book call it non directional because it can go to either direction.

Classical Approach

Calculate a test statistic, t or Z. Formulas for calculating t andZ are in the book.

Z test statistic is given on page 318.

T test statistic is given on page 328.

There are 3 possible tests:1. Right tailed test (described on pages 320 and 321)2. Left tailed test (similar to a right tailed test)3. Two tailed test (described on pages 318 and 319)

The flow chart on page 316 is a great help in guiding you onwhich method to use for any particular Hypothesis situation.

Difference Between T and Z tests

The only major difference to find a t value from the table t inthe back of the book, you need to take TWO things to the ttable.

One is alpha (that you already know about and is usuallygiven) and the other element is DEGREE of FREEDOM. Degreeof freedom is just a number that helps us to have a moreaccurate value for our t statistic.

DF (degree of freedom) value is sample size, n, minus 1 (n1). It is basically another factor that comes to play to bringaccuracy to the calculations based on different sample sizes.That is all there is into degree of freedom for us!

Example: t table

The T-table is located just before the Z-table on the insidecover of the book.

As book shows in the back of the book in Table t, if youare looking for a t value when alpha is 5% (one-sided or onetailed test) and sample size is 74, you go to the Table andlook up the t (0.05,73). The t value when sample size is 73and alpha is 0.05 is 1.666. Let me know if you are not gettingthis value from the t table in the back of the book right beforethe Z table.

T table

Just remember that the values in the body of the tablerepresent the shaded area (blue) in the t distribution as it isshown in the back of the book table t.

If sample size approaches the value of infinity then tdistribution approaches standard normal distribution and thetwo curves become identical. So, for example, Z of alpha0.05 = 1.645 which is the same value as t (df=infinity, alpha0.05) = 1.645.

In Practice

The t-test is the more practical case as we usually don't havethe standard deviation of a population parameter. If yourecall, when we know the standard deviation of the populationwe use Z test. Now, we use student t test (which has aformula which is very similar to Z test formula) becausestandard deviation of population is not known.

The only difference is that we use standard deviation of thesample, s, in the formula, instead of sigma. Calculation andconclusion of t test is very similar to the calculation andconclusion of a Z test. We can call these values -calculated tor Z

Example: t-test

A study of the process costs indicates that the average weightof the diamonds must be greater than 0.5 karat in order thatthe process be operated at a profitable level. Do the sixdiamond-weight measurements, 0.46, 0.61, 0.52, 0.48, 0.57,0.54 present sufficient evidence to indicate that the averageweight of the diamonds produces by the process is in excessof 0.5 karat?

We use t test because sample size is 6 (less than 30). It is aone-sided test because question is about the value being“greater than”.

H0: population average weight of the diamonds (mu) = 0.5H1: population average weight of the diamonds (mu) > 0.5

Example t-test

We decide that the value of alpha to be 0.05 (rejecting top5% of the t values). The degree of freedom is sample sizeminus 1 so degree of freedom (df) for this problem is 6-1 = 5.The Critical t value has the format of t alpha, df. So, for thisproblem, it is: t 0.05, 5= 2.015 (from t table in the back ofthe book).

That is, we will reject the Ho if the calculated t (calculatedusing the formula) is greater that maximum acceptable table twhich is 2.015 (for this problem). In that case, we say the calculated t is too large to be accepted according to our 5%policy.

Example t-test

So, the Rejection Region for alpha = 5% and (6-1)= 5degrees of freedom is when calculated t (using the formula) isgreater than 2.015 (look at the t distribution figure on the topof t –table in the back of the book. The red area is therejection area).

If you use the t formula for this problem you will findcalculated t value to be 1.31. In this case calculated t is lessthan critical t (table t), therefore, we do not reject the H0. This implies that the data do not present sufficient evidence toindicate that the mean diamond weight exceeds 0.5 karat.

P-Value Approach

The calculations, the meaning of alpha and P-value andconclusion process are the same in both methods butformulas are a little different. We will get familiar withgetting a t value from the table in our seminar a little lateron tonight.

Steps are outlined on pages 322 and 323.

p-value

The p-value is the probability in the “tail” area. In theclassical approach, you either reject of fail to reject. Itdoesn’t give any information about whether a different valueof alpha would have given the opposite conclusion.

In the p-value approach, you find the level of alpha at whichthe null hypothesis would be rejected. For example, if the pvalue is .2 and it is a one tailed test, it indicates that theprobability of getting a sample with the stated mean is twentypercent. (Pretty high and you would not want to reject thenull.) If the p-value is 0.0001, the probability that the sampleis drawn from a population stated in the null hypothesis is very, very small. (In this case, you would reject the null.)

test statistic value

p-value

The example shown is for a right tailed test.

test statistic value

p-value

The example shown is for a left tailed test.

test statistic value

p-value

The example shown is for a two tailed test. (Find the area inone tail, and double it.)

test statistic value