A.P. Statistics – t-tests - Important Concepts€¦ · Hypothesis Testing The procedure for hypothesis testing for t-tests is mostly the same as the z-test. The starred steps are

www.MasterMathMentor.com - 1 - Stu Schwartz

A.P. Statistics – t-tests - Important Concepts: The previous section taught you how to find confidence intervals and perform hypothesis tests and hopefully, you understand the procedures. The problem that we never addressed in that section is that all the problems were based on the unrealistic assumption that we knew the population standard deviation

!

" . Our knowing that value is highly improbable and yet, knowing it allowed us to use z-tests and z-confidence intervals. In this section, we are much more realistic and do not depend on

!

" . We are still going to sample via an SRS from a population with mean

!

µ and standard deviation

!

" . Since we don’t know

!

" , we are going to estimate it. When we sample from a population, we will get a mean

!

x and a standard deviation s. We estimate the standard deviation of

!

x by using

!

s n .

We define standard error of the sample mean

!

x to be

!

s

n

When we knew the value of

!

" , we found our z-statistic by the formula

!

z =x "µ

0

#

n

. Now that we don’t know

!

"

and are going to estimate it using our standard error

!

s

n, we create a new statistic called the t-statistic.

Draw an SRS of size n from a population with unknown

!

µ and standard deviation

!

" .

The one-sample t-statistic is defined as

!

t =x "µ

0

s

n

.

The t-distribution has n – 1 degrees of freedom (which we will discuss shortly). The t-distribution looks very much like the normal distribution (for which we use z) that follows the 68-95-99.7 rule. However, since we do not know

!

" , we had to estimate it by using n. The larger n is, the closer our t-distribution looks like the z (normal) distribution. We describe this with n – 1 degrees of freedom which is abbreviated as (df). The picture on the right shows 3 curves. The highest curve is the standard normal (z) distribution. The innermost curve is the t distribution with 2 degrees of freedom and the middle curve is the t-distribution with 6 degrees of freedom. Note that the t-distribution shapes are symmetric but have greater probability in the tails than the normal distribution. Also note that has the number of degrees of freedom increases, the t-distribution approaches the normal distribution. This happens because s estimates

!

" more accurately as the sample size n increases. So using s in place of

!

" causes little problem when the sample is large.


So, rather than going through our procedures for confidence intervals and hypothesis testing using t rather than z, I am going to lay it all out for you on this page and then we will do a number of examples illustrating them. However, remember, for right now, you decide on z vs. t based on one criterion: are you given

!

" ? If you are not (which is usual), you use the t-procedures outlined below. Draw an SRS of size n from a population with unknown mean

!

µ.

Confidence Intervals

To find the level C confidence interval, use the formula:

!

x ± t *s

n

"

# $

%

& ' with n – 1 degrees of freedom (df).

!

t *s

n is your margin of error.

To find t*, use the t-distribution table at the back of the book. Go to the row for your degrees of freedom (df) … that is n – 1. Now, move to the column for your desired confidence level C – those numbers are at the bottom. The intersection of the correct row and column is your t*. Note that if n is above 30, the rows skip 10. It won’t matter much whether you use the lower or upper row as these values are very close.

Hypothesis Testing

The procedure for hypothesis testing for t-tests is mostly the same as the z-test. The starred steps are different. 1) describe your parameter of interest

!

µ 2) describe the test you are using – in this case the t-test because you are not given

!

" . 3) you will need, n,

!

x , and s. * 4) Check your conditions for the test. Those are described below. 5) Write your null hypothesis

!

H0:µ = µ

0 and alternative hypothesis

!

Ha using variables – either a left, right,

or 2-tailed test.

* 6) Find your test statistic t using the formula:

!

t =x "µ

0

s n. The degrees of freedom (df = n – 1) is reported.

* 7) To find your p-value, use the t-table.. Choose the row for your degree of freedom (df) which will be row n – 1. Look across that row until you find the t value you just computed. You most probably won’t find it. So find the one it is closest to and then read up to the top row – upper tail probability. That is your p-value. It won’t be exact, but it isn’t necessary for complete accuracy. 8) Decide on your

!

" . As before, if it isn’t specified, use

!

" = .05 . Compare it to your p-value. 9) If your p-value <

!

" , the results are significant. You reject

!

H0 and accept

!

Ha. If p-value >

!

" , then the results are not significant and you fail to reject

!

H0.

10) Make a final conclusion in English remembering to use the suggested language given using z-tests.


Conditions for t-tests or t-Confidence Intervals

• We always assume an SRS. If it is not clearly stated in the problem, it should be mentioned as assumed.

• Sample size n < 15. Use t-procedures if the data in the sample appears normal. If the data is clearly non-normal or there are outliers, do not use t-procedures.

• Sample size < 30. It is safe to use t-procedures unless there is a very strong skew. • Sample size ≥ 30. It is safe to use t-procedures even if there are outliers or strong skew. When you do a hypothesis test or confidence interval using t, you should note whether the conditions are met. If they are not, you should proceed with the test anyway and then make mention of your concern about the accuracy of the test.

Example: SRS assumed with n = 20, data appears normal. or SRS, n = 9, outlier present. The result of the hypothesis test is open to question.

A confidence interval or significance test is called robust if the confidence interval or p-value does not change very much when the assumptions of the procedure are violated. When we do confidence intervals, you must mention the conditions as well. It is a form of hypothesis test. With all of this knowledge, we can now do problems # 1, 2 and 3 on pages 7 and 8. Comparing Two Means We have examined and performed hypothesis tests with the null hypothesis

!

H0:µ = µ

0 against some alternative

hypothesis. We question if there is truly an average of 80 fries in the packet, whether there is evidence that gasoline purchases truly average 10 gallons, whether this class can throw paper airplanes 20 feet. In all cases, you are testing against some specific number. They are called one-sample tests, whether z or t. But it is possible to do two-sample tests as well – to compare one set of data to another. • Is one drug more effective than another? • Do girls have more social insight than boys? • Does one incentive plan promote the use • Do students do better in courses when calculators of one credit card over another? are allowed compared to when they are not? • Do people prefer white or dark chocolate? • Does sunlight promote growth of plants? The first thing to remember when comparing two sets of data is to realize that even when you are given two sets, hypotheses tests and confidence intervals must still be done by the one sample t procedures above if the data comes from a matched pairs design. Matched Pairs data comes from an experiment in which the data is not independent. In a matched pairs design, one set of data is matched with the other. There will be exactly the same amount of data in one data set as the other. In a matched pairs design, you take the two data and subtract one from the other. The typical hypothesis test will now compare the difference in data to zero or some other number to determine if the treatment made a difference. The typical confidence interval will look at the difference between the two treatments.


Typical matched pairs experiments typically look like: Does taking a medication lower blood pressure? • Collect data …. Give treatment … collect data again. Example: take blood pressure of subjects … give medication … take blood pressure again

BP before med BP after med BP Difference Subject 1 135 125 10 Subject 2 132 130 2 Subject 3 126 129 -3

!

µ = Difference in BP after taking medication After " Before( )

H0 : µ = 0

Ha

: µ < 0

An observational study using a matched pairs design would examine two sets of data from the same subject. For instance, if we were interested if there is a difference how people like Pepsi as opposed to Coke, we would run a blind taste test. Example: Subject rates Pepsi in terms of taste and rates Coke in terms of taste

Pepsi rating Coke rating Difference Subject 1 7 5 2 Subject 2 8 8 0 Subject 3 2 6 -4

!

µ = Difference in Pepsi rating and Coke rating

H0 : µ = 0

Ha

: µ " 0

With this knowledge, you can now do problem # 4 on page 9. Two Sample Problems Comparing two populations or two treatments is one of the most common situations encountered in everyday statistical practices. These situations are called two-sample problems. In a two-sample problem, the goal of inference is to compare the response to two treatments or compare the characteristics of two populations. We will have a separate sample from each treatment or each population. Again, repeating some typical two-sample problems: • Is one drug more effective than another? • Do girls have more social insight than boys? • Does one incentive plan promote the use • Do students do better in courses when calculators of one credit card over another? are allowed compared to when they are not? • Do people prefer white or dark chocolate? • Does sunlight promote growth of plants?


Again, remember you must decide whether the experiment is a true two-sample experiment or a matched pairs design. The first problem, comparing drugs is most probably two-sample. You give drug A to one set of subjects and drug B to another set of subjects and compare the results. In the chocolate problem, you have a choice, You could give one set of people chocolate A to taste and another set of people chocolate B to taste, take the average rating and compare them. That would be a two-sample design. However, you could give each subject both chocolates to taste, get the ratings for both, and look at the difference between the ratings. That would be a one-sample matched pairs design. So, when we have a two-sample design, the conditions are these:

• We have two SRS’s from two distinct populations. The samples are independent. Matched pairs violates independence and would then be done with a one-sample procedure.

• Both populations are normally distributed or

!

n1 and n

2" 30. These must be mentioned or verified

when you do your two-sample confidence interval or hypothesis test.

Confidence Intervals for Two-samples

We have two independent samples: sample 1 with unknown population mean

!

µ1,n

1,x

1 and s

1

sample 2 with unknown population mean

!

µ2,n

2,x

2 and s

2

To find the level C confidence interval for

!

µ1"µ

2, use the formula:

!

x 1" x

2( ) ± t *s1

2

n1

+s2

2

n2

The degrees of freedom (df) will be the smaller of

!

n1"1 and

!

n2"1

!

t *s1

2

n1

+s2

2

n2

is your margin of error.

To find t*, use the t-distribution table at the back of the book. Go to the row for your degrees of freedom (df) … that is the smaller of

!

n1"1 and

!

n2"1. Now, move to the column for your desired

confidence level C – those numbers are at the bottom. The intersection of the correct row and column is your t*. Note that if n is above 30, the rows skip 10. It won’t matter much whether you use the lower or upper row as these values are very close.

You may wonder where

!

t *s1

2

n1

+s2

2

n2

comes from and why it is so much more complicated that the formula we

used in one sample t-confidence intervals:

!

t *s

n. Recall from chapter 7, to find the standard deviation of the

sum of two random variables, we had to find the variances of both by squaring them, adding them, and then taking the square roots. The same thinking goes here.


Hypothesis Testing The procedure for hypothesis testing for two-sample t-tests is mostly the same as the one-sample t-test. The starred steps are different. Your conditions are not the same and your t-statistic is found differently. But you still create null and alternative hypotheses, generate that t-statistic, and compare it to your

!

" level. * 1) describe your parameter of interest. In this case, it will be

!

µ1"µ

2, the difference between the two

population means 2) describe the test you are using – in this case the two-sample t-test. 3) you will need,

!

n1,x

1,s

1,n

2,x

2 and s

2

* 4) Check your conditions for the test: Both populations are normally distributed or

!

n1 and n

2" 30. Also, both

samples are from an SRS. 5) Write your null hypothesis

!

H0

: µ1

= µ2 or H

0: µ

1"µ

2= 0 and alternative hypothesis

!

Ha using variables –

either a left, right, or 2-tailed test.

* 6) Find your test statistic t using the formula:

!

t =x 1" x

2

s1

2

n1

+s2

2

n2

. The degrees of freedom (df) is reported.

7) To find your p-value, you will use table A. Choose the row for your degree of freedom (df) which will be row n – 1. Look across that row until you find the t value you just computed. You most probably won’t find it. So find the one it is closest to and then read up to the top row – upper tail probability. That is your p-value. It won’t be exact, but it isn’t necessary for complete accuracy. 8) Decide on your

!

" . As before, if it isn’t specified, use

!

" = .05 . Compare it to your p-value. 9) If your p-value <

!

" , the results are significant. You reject

!

H0 and accept

!

Ha. If p-value >

!

" , then the results are not significant and you fail to reject

!

H0.

10) Make a final conclusion in English remembering to use the suggested language given previously. You know have enough information to complete problems 5 and 6 on page 10.


Chapter 11 – t-test example problems

Instructor’s note: This and succeeding problems comes from The Practice of Statistics, (Yates, Moore, Starnes), chapter 11 in which students make and fly fixed wing planes and sleek jets. If you have enough students in your class(es), it is recommended that they make the planes, fly them, and you collect the data in terms of how far they fly (feet). If not, then use the data provided below which came from a statistics class. Fixed Jet Gender Fixed 12.1 6.4 8.3 7.2 8.2 3.6 4.7 7.4 8.6 8.9 5.8 16.5 6.9 4.9 8 11.7 14.3 9.1 7.6 Jet 19 13.5 6.7 5 12.6 4.2 12.6 8 12.4 11.4 20 15.3 16.5 13.5 7.7 8.2 11.9 12.8 14.5 Gender B G G B B G B G G B B B G G G B B B G Calculate the following variables:

!

x F

= ____ sF

= ____ nF

= _____ x J

= ____ sF

= ____ nJ

= _____

1. Over the years I have done this experiment, I have found that students were able to throw the fixed wing plane an average of 7 feet. Is there evidence that this statistics class can throw it farther than 7 feet. Explain why or why not you can make that claim.

2. Assume that this statistics class is a random sample of students with paper plane making and flying ability from this school. Is there evidence that students in this school are better at flying fixed planes that the previous classes? Show all work. Also, find and interpret the 95% confidence interval for the fixed wing data.

Test: Given information: Parameter of interest (appropriate to the problem situation): Conditions: Null and Alternative Hypotheses: Appropriate statistic (either by shown formula or calculator – underline) p-value ___________________ Conclusion regarding Ho: ___________________________________________ Conclusion for problem (in clear English): 95% Confidence interval (either by shown formula or calculator – underline) _____________________ Meaning of confidence interval in words:


3. Over the years I have done this experiment at other schools, I have found that students were able to throw sleek jets an average of 10 feet. My observation is that this school is different in many respects to previous schools so I believe that the student’s abilities to fold and fly the sleek jets are different as well. Again, using this class as a random sample of jet folders and flyers in this school, is there evidence to support my conjecture? Also, find and interpret the 95% confidence interval for the sleek jet wing data.



4. Again, using this class as a random sample from this school, is there evidence to support the conjecture that the sleek jets fly farther than the fixed wing planes. Show all steps. Also find and interpret the 95% confidence interval for the difference between the sleek jet and fixed wing data.



5. Everyone knows (!) that boys are better at folding and flying paper airplanes than girls, especially more complicated ones. Comment on and give evidence whether that statement is true for this class.

6. Assuming that this class is a random sample of sleek jet folders and flyers in this school, decide on whether there is enough evidence that boys are better sleek jet builders and flyers than girls. Again, show all steps. Then create and interpret a 95% confidence interval on the difference between boys and girls in sleek jet flying.



Using the Calculator As we saw with z-tests and confidence intervals, t-tests and t-intervals, both one and two-sample, are built into the calculator. All are found in the

!

STAT EDIT menu. As before, our tests can either use

summary statistics

!

Stats or actual data

!

Data . For the problems just given, we will of course use data and below are the screens that you will get for each problem. We start off by putting out data into the calculator. I have decided to put the fixed-wind data into L1 and the sleek jet data into L2. Problem 2: We want to see if there is evidence that our fixed wing data (L1) is greater than 7. This is a t-test (we do not know the population standard deviation). To the right are the screens. The important information is the fact that t = 1.915 and the p-value is .036, which at the 5% level is enough to reject the null hypotheses

!

H0:µ

F= 7 and say that there is enough evidence that students from this school

can throw fixed-wing planes greater than 7 feet.

If we wish to find a confidence interval, we use a TInterval. The results say that we are 95% confident that students from this school can fly fixed-wing planes on an average of between 6.86 and 10.00 feet. Problem 3: We are interested in whether there is enough evidence that this school flies jets different from the other class who flew them on average 10 feet. The differences between this and problem 2 is that we are now using the sleek jet data in L2 and our test will be a two-tailed test. The screens are to the right: In this one, I showed the graph as well. The important information is that t = 1.880 and the p-value is .076 which at the 5% level does not give enough evidence to reject the null hypothesis

!

H0:µ

J=10 . Your statement is that

there is not enough evidence that students from this school throws sleek jets different from the other class. If we wish to find a confidence interval, we still use a TInterval. The results say that we are 95% confident that students from this school can fly sleek jets on an average of 9.78 and 13.99 feet. Notice how 10 feet is included in that interval which is why we didn’t reject the null hypothesis above.


Problem 4: We are interested in whether there is enough evidence that this school’s students fly sleek jets further than fixed wing planes. Though there are two samples, they are not independent as the plane was made by and flown by the same person. So we need to look at the difference between the data. We will put that in L3. We now want to test whether there is evidence that the difference (L3) is positive so our null hypothesis is

!

H0:µ

difference= 0. We treat this as a one-sample t-test as we did in problem 2 above.

The important information is that

!

x = 3.45 , t = 3.08 and the p-value is .003. This is strong evidence to reject

!

H0 and claim that students from this school throw jets further than fixed wings.

If we wish to find a confidence interval, we still use a TInterval. The results say that we are 95% confident that students in this school will throw sleek jets from 1.095 to 5.81 feet further than fixed wings. Notice how zero is not in that interval which is why we rejected the null hypothesis above.

Problem 6: We are interested in whether there is enough evidence that boys in this school flies sleek jets further than girls so. Since we only are interested in jet data and we want to distinguish between boys and girls. we are going to have to change our lists. In L1, we want the Boy Jet data and in L2 we want the Girl Jet data. In this case, we want a two-sample t-test because they are two independent samples. We now wish to test whether the mean of boys (in L1) is greater than the mean of girls (in L2). So our null hypothesis is:

!

H0 : µboy "µgirl = 0 or H0 : µboy = µgirl . In the test screen you are asked whether you wish Pooled data. The answer to this is NO. (This is an optional topic). The important information needs two screens to display (you need to scroll). Note that you are given the t-statistic = 1.054 and the p-value = .153. The values of

!

x 1 and

!

x 2 (the boy and girl average) along

with their st. deviations and their values of n (how many in each sample). The fact that your p-value is .153 shows that there is not enough evidence to reject the null hypothesis so we conclude that there is no evidence that boys in the school throw jets further than girls. Above, you were told that the degrees of freedom was the smaller of

!

n1"1 and

!

n2"1 which are integers.

That is not what the calculator displays. The calculator uses a different and more complex formula to calculate the degree of freedom. It does so with more accuracy than the simple method just described. Whichever method you use, it rarely will make a difference in the p-value which is the result of the significance test. If you do the problem using your tables, report the integer df. If you use the calculator, report the df that the calculator uses. As said, it rarely will make a difference. Finally, if you want the confidence interval, you want the 2-Sample TInterval. The screen is set up the same way it was above (no pooling). The results show that we are 95% confident that the difference between boys and girls is -2.105 to 6.309. The fact that the lower bound is negative shows that it is quite possible that girls throw jets further than boys.


Homework Problems

1. The builders of condominiums claim that the total room area of the condos is1,500 ft2. A random sample of 35 condos shows that the average total area is 1,488 ft2 with a standard deviation of 24.25 ft2. Is there evidence at the 5% level that the condos have less area than the advertised number? Also find a 95% confidence interval for the size of the condos.



2. The maker of a hybrid car claims that the car will get 40 miles to the gallon (mpg). A random sample of 20 cars were tested to see if the mileage is not as high as claimed. Perform a significance test at the 5% level and find a 95% confidence level for the mileage of the hybrid car.

37.2 38.4 38.9 40.0 42.3 39.7 37.8 39.2 40.8 39.9 37.9 39.2 40.4 40.0 38.7 39.8 42.8 41.1 38.9 39.5



3. A Las Vegas casino advertises its slot machines in a certain area of the casino as an average of 95% payback. An SRS of 15 slot machines were tested to see if the average payback was different than 95%. Perform a hypothesis test at the 2% level and then generate a 98% confidence interval for the average payback of a slot machine in this area. The paybacks are below.

.889 .901 .911 .908 .961 .912 .923 .899 .925 .955 .987 .931 .920 .911 .931



4. The Macintosh Leopard operating system is advertised to be faster than the PC Vista operating system. Non-biased testers chose 12 random tasks and timed them. Is there evidence at the 5% level that Leopard is faster than Vista? Also, find a 95% confidence interval for the difference in time between the two operating systems. The times are given below in seconds.

Task A B C D E F G H I J K L Leopard 12.5 29.3 9.1 24.4 19.5 28.1 3.6 39.4 45.9 28.9 17.3 50.0 Vista 11.3 32.4 9.3 30.6 22.2 28.0 3.9 42.5 55.1 31.3 14.4 53.3



5. Teeth whitening is now big business. Dentists use a numbering system to determine the whiteness of teeth. Pure white is 10 and very dark (yellowed) teeth is 0. Two whitening systems are tested, All-White and Bright Smile. All-White uses a liquid while Bright Smile uses strips. A random sample of people with dark teeth (10’s) are chosen and given one of these systems for a period of two months. At the end of that time, their teeth were tested for whiteness. Is there evidence that All-White whitens teeth better than Bright Smile. Also find a 95% confidence interval between All-White and Bright Smile. Statistics are below.

!

x s n All-White 6.42 0.81 24 Bright Smile 6.15 0.49 18



6. Some universities believe that extended use of graphing calculators make students less proficient in tasks for which graphing calculators are not permitted (they make students lazy). Two classes were taught with the same instructor, one making use of graphing calculators while the other not using them. At the end of the course, an exam was given where calculators were not permitted. Is there evidence at the 5% level that the students who were not allowed graphing calculators in the course did better than those who were allowed the calculators. Also find the 95% confidence interval for the difference.

Graphing Calculator course Non Graphing Calculator

82 84 71 82 93 92 83 78 93 79 71 68 72 74 83 82 88 87 79 80 82 91 93 85 75 75 83 99 66 83 77 76 84 91 86 86 92 93 90 84 69 70 74 67 82 88



7. A new cold medicine claims to reduce the amount of time a typical person “suffers” with a cold. A random sample of 25 people were chosen for an experiment which consisted of taking the new medication when they felt the onset of a cold and continue to take it twice a day until they “feel better.” The average time these people took the medication was 6.2 days with a standard deviation of 1.4 days. The typical time a person suffers with a cold is said to be one week. Do an appropriate hypothesis test at the 1% level to determine if the new medication seems to work. Also, construct and interpret a 99% confidence interval for this data.



8. It is thought that people paying small purchases (under $25) in a bookstore with cash will spend less than people paying with credit cards. On a Saturday, a random sample of 20 customers who made purchases under $25 and who paid with cash were selected and their purchases averaged. They spent on average $15.45 with a standard deviation of $3.64. The typical credit card purchases (under $25) is $17.30. Do an appropriate hypothesis test to check out the original conjecture as well as construct and interpret a 95% confidence interval.



9. Traffic experts believe that placing an electronic sign tell drivers of their current speeds will generally reduce speeds. On a busy road, such a sign was placed and a random sample of 20 drivers had their speeds recorded. The sign was taken down and on a day with similar weather conditions, 20 random drivers had their speeds recorded. Do an appropriate hypothesis test at the 5% level to determine if the traffic expert’s beliefs are well founded and then construct and interpret a 98% confidence interval for the data.

Without Sign

38 52 55 46 48 52 65 38 40 42 53 50 50 29 48 37 65 48 52 49

With sign

40 50 50 60 35 44 25 37 55 40 38 30 55 45 38 40 55 50 35 22



10. It is thought that listening to the sound of the surf will help people get to sleep faster. An experiment was done using 10 people. Some nights the sound of surf was piped into their bedroom and other nights, no sound was used. The experiment was run for a period of time and the average time it took for them to fall asleep was monitored and averaged. Following is the data for the 10 people. Perform a hypothesis test to determine the possible effectiveness of the sound of the surf for sleep and construct a 95% confidence interval for the difference in time it takes to fall asleep without the sound of the surf as opposed to with it.

With surf

10.2 12.5 13.2 23.1 7.5 8.9 12.2 13.4 18.4 17.5

Without surf

11.4 13.4 13.2 22.3 8.8 12.3 14.2 13.3 19.9 19.7


Documents

A.P. Statistics – t-tests - Important Concepts€¦ · Hypothesis Testing The procedure for hypothesis testing for t-tests is mostly the same as the z-test. The starred steps are