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SECTION 12.1 SECTION 12.1 Tests About a Population Tests About a Population Mean Mean

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SECTION 12.1. Tests About a Population Mean. What’s the difference between what is addressed in Section 11.2 (we skipped) and what we are beginning in Section 12.1?. - PowerPoint PPT Presentation

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Page 1: SECTION 12.1

SECTION 12.1 SECTION 12.1

Tests About a Population MeanTests About a Population Mean

Page 2: SECTION 12.1

What’s the difference between What’s the difference between what is addressed in Section what is addressed in Section

11.2 (we skipped) and what we 11.2 (we skipped) and what we are beginning in Section 12.1?are beginning in Section 12.1?

In reality, the standard deviation In reality, the standard deviation σσ of the of the population is unknown, so the procedures population is unknown, so the procedures from last chapter are not useful. However, from last chapter are not useful. However, the understanding of the logic of the the understanding of the logic of the procedures will continue to be of use. procedures will continue to be of use.

In order to be more realistic, In order to be more realistic, σσ is estimated from is estimated from the data collected using sthe data collected using s

Page 3: SECTION 12.1

Overview of a Significance TestOverview of a Significance Test A A test of significancetest of significance is intended to assess the evidence is intended to assess the evidence

provided by data against a provided by data against a null hypothesis null hypothesis HH0 0 in favor of in favor of an alternate hypothesis Han alternate hypothesis Haa..

The statement being tested in a test of significance is The statement being tested in a test of significance is called the called the null hypothesisnull hypothesis. Usually the null hypothesis is . Usually the null hypothesis is a statement of “no effect” or “no difference.” a statement of “no effect” or “no difference.”

A A one-sided one-sided alternate hypothesis exists when we are alternate hypothesis exists when we are interested only in deviations from the null hypothesis in interested only in deviations from the null hypothesis in one directionone direction

HH00 : : =0 =0 HHa a : : >0 (or >0 (or <0)<0)

If the problem does not specify the direction of the If the problem does not specify the direction of the difference, the alternate hypothesis is difference, the alternate hypothesis is two-sidedtwo-sided

HH00: : =0=0HHaa: : ≠≠00

Page 4: SECTION 12.1

HYPOTHESESHYPOTHESESNOTE: Hypotheses ALWAYS NOTE: Hypotheses ALWAYS

refer to a population parameter, refer to a population parameter, not a sample statistic.not a sample statistic.

The alternative hypothesis should The alternative hypothesis should express the hopes or suspicions express the hopes or suspicions we have BEFORE we see the we have BEFORE we see the data. Don’t “cheat” by looking at data. Don’t “cheat” by looking at the data first.the data first.

Page 5: SECTION 12.1

CONDITIONSCONDITIONS These should be VERY FAMILIAR to you by now.These should be VERY FAMILIAR to you by now.

RandomRandomData is from an SRS or from a randomized experimentData is from an SRS or from a randomized experiment

NormalNormalFor means—population distribution is Normal or you have a large sample For means—population distribution is Normal or you have a large sample

size (nsize (n≥30) to ensure a Normal sampling distribution for the sample mean≥30) to ensure a Normal sampling distribution for the sample meanFor proportions—np≥10 and n(1-p)≥10 (meaning the sample is large For proportions—np≥10 and n(1-p)≥10 (meaning the sample is large

enough to ensure a Normal sampling distribution for )—more details in enough to ensure a Normal sampling distribution for )—more details in next Section 12.2next Section 12.2

IndependentIndependentEither you are sampling with replacement or you have a population at Either you are sampling with replacement or you have a population at

least 10 times as big as the sample to make using the formula for st. least 10 times as big as the sample to make using the formula for st. dev. okay. dev. okay.

Page 6: SECTION 12.1

CAUTIONCAUTIONBe sure to check that the conditions Be sure to check that the conditions

for running a significance test for for running a significance test for the population mean are satisfied the population mean are satisfied before you perform any before you perform any calculations.calculations.

Page 7: SECTION 12.1

t Statistict Statisticnote: this is the same as what we learned in Chapter 10note: this is the same as what we learned in Chapter 10

The statistic does not have a normal distributionThe statistic does not have a normal distribution Degrees of freedom: n-1Degrees of freedom: n-1 Differs from a z statistic because Differs from a z statistic because σσ is not used is not used t statistic says how far x is from its mean t statistic says how far x is from its mean μμ in in

standard deviation unitsstandard deviation units We are now using the body of the table we used in We are now using the body of the table we used in

the last chapter.the last chapter.

0xt

sn

Page 8: SECTION 12.1

ROBUSTNESSROBUSTNESS

ROBUST: ROBUST: Confidence levels or Confidence levels or PP-Values -Values do not change when some of the do not change when some of the assumptions are violated assumptions are violated

Fortunately for us, the t-procedures are Fortunately for us, the t-procedures are robust in certain situations.robust in certain situations.

Therefore . . . Therefore . . .

Page 9: SECTION 12.1

This is when we use the t-proceduresThis is when we use the t-procedures It’s more important for the data to be It’s more important for the data to be

an SRS from a population than the population an SRS from a population than the population has a normal distributionhas a normal distribution

If n is less than 15, the data must be normal to If n is less than 15, the data must be normal to use t-proceduresuse t-procedures

If n is at least 15, the t-procedures can be used If n is at least 15, the t-procedures can be used except if there are outliers or strong skewnessexcept if there are outliers or strong skewness

If n≥40, t-procedures can be used even in the If n≥40, t-procedures can be used even in the

presence of strong skewnesspresence of strong skewness

Page 10: SECTION 12.1

Density Curves for Density Curves for t Distributionst Distributions

Bell-shaped and symmetricBell-shaped and symmetricGreater spread than a normal curveGreater spread than a normal curveAs degrees of freedom (or sample size) As degrees of freedom (or sample size)

increases, the t density curves appear increases, the t density curves appear more like a normal curvemore like a normal curve

Page 11: SECTION 12.1

Tip on Interpreting Tip on Interpreting PP-values-values On page 746, at the bottom of the box “The One-On page 746, at the bottom of the box “The One-

Sample t Test,” it is stated, “These Sample t Test,” it is stated, “These PP-values are -values are exact if the population distribution is Normal and exact if the population distribution is Normal and are approximately correct for large n in other are approximately correct for large n in other cases.” Example 12.2 (same page) then uses a cases.” Example 12.2 (same page) then uses a small sample (n=10) with no guarantee that the small sample (n=10) with no guarantee that the distribution is Normal and where the Normal distribution is Normal and where the Normal probability plot is a bit “iffy.” Thus, this is a case in probability plot is a bit “iffy.” Thus, this is a case in which we choose to use t where we assume that which we choose to use t where we assume that the population distribution is approximately Normal the population distribution is approximately Normal because we don’t have clear evidence that it isn’t. because we don’t have clear evidence that it isn’t. Therefore, the Therefore, the PP-values are approximately correct.-values are approximately correct.

Page 12: SECTION 12.1

INFERENCE TOOLBOX (p 705)INFERENCE TOOLBOX (p 705)

1—1—PPARAMETER—Identify the population of interest ARAMETER—Identify the population of interest and the parameter you want to draw a conclusion and the parameter you want to draw a conclusion about. STATE YOUR HYPOTHESES!about. STATE YOUR HYPOTHESES!

2—2—CCONDITIONS—Choose the appropriate inference ONDITIONS—Choose the appropriate inference procedure. VERIFY conditions (procedure. VERIFY conditions (Random, Normal, Random, Normal, Independent) Independent) before using it.before using it.

3—3—CCALCULATIONS—If the conditions are met, carry ALCULATIONS—If the conditions are met, carry out the inference procedure.out the inference procedure.

4—4—IINTERPRETATION—Interpret your results in the NTERPRETATION—Interpret your results in the context of the problem. CONCLUSION, context of the problem. CONCLUSION, CONNECTION, CONTEXT(meaning that our CONNECTION, CONTEXT(meaning that our conclusion about the parameter connects to our work conclusion about the parameter connects to our work in part 3 and includes appropriate context)in part 3 and includes appropriate context)

Steps for completing a SIGNIFICANCE TEST:

DO YOU REMEMBER WHAT THE STEPS ARE???

Page 13: SECTION 12.1

Step 1—PARAMETERStep 1—PARAMETER

Read through the problem and determine Read through the problem and determine what we hope to show through our test.what we hope to show through our test.

Our null hypothesis is that no change has Our null hypothesis is that no change has occurred or that no difference is evident.occurred or that no difference is evident.

Our alternative hypothesis can be either Our alternative hypothesis can be either one or two sided.one or two sided.

Be certain to use appropriate symbols and Be certain to use appropriate symbols and also write them out in words.also write them out in words.

Page 14: SECTION 12.1

Step 2—CONDITIONSStep 2—CONDITIONS Based on the given information, determine which test Based on the given information, determine which test

should be used. Name the procedure.should be used. Name the procedure. State the conditions.State the conditions. Verify (through discussion) whether the conditions Verify (through discussion) whether the conditions

have been met. For any assumptions that seem have been met. For any assumptions that seem unsafe to verify as met, explain why. Don’t forget, with unsafe to verify as met, explain why. Don’t forget, with the t distribution, there is more “forgiveness” due to the t distribution, there is more “forgiveness” due to the robustness of the t proceduresthe robustness of the t procedures

Remember, if data is given, graph it to help facilitate Remember, if data is given, graph it to help facilitate this discussionthis discussion

For each procedure there are several things that we For each procedure there are several things that we are assuming are true that allow these procedures to are assuming are true that allow these procedures to produce meaningful results.produce meaningful results.

Page 15: SECTION 12.1

Step 3—CALCULATIONSStep 3—CALCULATIONS

First write out the formula for the test statistic, First write out the formula for the test statistic, report its value, mark the value on the curve.report its value, mark the value on the curve.

Sketch the density curve as clearly as possible Sketch the density curve as clearly as possible out to three standard deviations on each side.out to three standard deviations on each side.

Mark the null hypothesis and sample statistic Mark the null hypothesis and sample statistic clearly on the curve.clearly on the curve.

Calculate and report the P-valueCalculate and report the P-valueShade the appropriate region of the curve.Shade the appropriate region of the curve.Report other values of importance (standard Report other values of importance (standard

deviation, df, critical value, etc.)deviation, df, critical value, etc.)

Page 16: SECTION 12.1

Step 4—INTERPRETATIONStep 4—INTERPRETATION There are really two parts to this step: decision & There are really two parts to this step: decision &

conclusion. TWO UNIQUE SENTENCES.conclusion. TWO UNIQUE SENTENCES. Based on the Based on the PP-value, make a decision. Will you -value, make a decision. Will you

reject Hreject H00 or fail to reject H or fail to reject H00.. If there is a predetermined significance level, then If there is a predetermined significance level, then

make reference to this as part of your decision. If not, make reference to this as part of your decision. If not, interpret the interpret the PP-value appropriately.-value appropriately.

Now that you have made a decision, state a Now that you have made a decision, state a conclusion IN THE CONTEXT of the problem.conclusion IN THE CONTEXT of the problem.

This does not need to, and probably should not, have This does not need to, and probably should not, have statistical terminology involved. DO NOT use the statistical terminology involved. DO NOT use the word “prove” in this statement.word “prove” in this statement.

Page 17: SECTION 12.1

The Steps for a The Steps for a ONE SAMPLE t-TESTONE SAMPLE t-TESTSame Approach—Slightly Different LookSame Approach—Slightly Different Look

1.1. State the hypotheses and name testState the hypotheses and name testHHoo: : = 0 = 0

HHaa: : ‹, ›, or ≠ 0 ‹, ›, or ≠ 0

2.2. State and verify your assumptionsState and verify your assumptions3.3. Calculate the Calculate the P-P-value and other important valuesvalue and other important values

- Done in calculator or…Done in calculator or…- Book:Book:

- Using Table C, look in the df (n-1) column and then look Using Table C, look in the df (n-1) column and then look across the line to find the range of probabilities the t statistic across the line to find the range of probabilities the t statistic falls in falls in

4.4. State Conclusions State Conclusions (Both statistically and contextually)(Both statistically and contextually)

- The smaller the - The smaller the PP-value, the greater the evidence is to -value, the greater the evidence is to reject Hreject Hoo

STATE

PLAN

DO

CONCLUDE

Page 18: SECTION 12.1

Summarizing the STEPS of InferenceSummarizing the STEPS of Inference

State the null and alternative hypotheses State the null and alternative hypotheses in contextin context

Identify the inference procedure to be Identify the inference procedure to be used and justify the conditions for its useused and justify the conditions for its use

Perform statistical mechanicsPerform statistical mechanicsState the conclusion in the context of the State the conclusion in the context of the

problem with a clear linkage to the problem with a clear linkage to the mechanics that imply that conclusionmechanics that imply that conclusion

Page 19: SECTION 12.1

Example 1-sided t-TestExample 1-sided t-Test The diastolic blood pressure for American women The diastolic blood pressure for American women

aged 18-44 has approximately the Normal distribution aged 18-44 has approximately the Normal distribution with mean with mean =75 milliliters of mercury (mL Hg) and =75 milliliters of mercury (mL Hg) and standard deviation standard deviation s=10 mL Hg. We suspect that s=10 mL Hg. We suspect that regular exercise will lower blood pressure. A regular exercise will lower blood pressure. A random sample of 25 women who jog at least random sample of 25 women who jog at least five miles a week gives sample mean blood five miles a week gives sample mean blood pressure pressure =71 mL Hg. Is this good evidence =71 mL Hg. Is this good evidence that the mean diastolic blood pressure for the that the mean diastolic blood pressure for the population of regular exercisers is lower than 75 population of regular exercisers is lower than 75 mL Hg?mL Hg?

x

Page 20: SECTION 12.1

Step 1Step 1The parameter of interest is the mean diastolic The parameter of interest is the mean diastolic

blood pressure blood pressure ..Our null hypothesis is that the blood pressure is Our null hypothesis is that the blood pressure is

no different for those that exercise.no different for those that exercise.Our alternative hypothesis is one-sided Our alternative hypothesis is one-sided

because we suspect that exercisers have lower because we suspect that exercisers have lower blood pressure. blood pressure.

HH00: : = 75 mL = 75 mL

HHaa: : < 75 mL < 75 mL

Page 21: SECTION 12.1

Step 2Step 2 Since we do not know the population standard Since we do not know the population standard

deviation we will be performing a t-test of significance. deviation we will be performing a t-test of significance. We were told that the sample is random, but we do not We were told that the sample is random, but we do not

know if it is an SRS from the population of interest. know if it is an SRS from the population of interest. This may limit our ability to generalize.This may limit our ability to generalize.

Since the population distribution is approximately Since the population distribution is approximately Normal, we know that the sampling distribution of Normal, we know that the sampling distribution of will also be approximately Normal. So we are safe will also be approximately Normal. So we are safe using the t procedures.using the t procedures.

The blood pressure measurements for the 25 joggers The blood pressure measurements for the 25 joggers should be independent. Note that the population of should be independent. Note that the population of interest is at least 10 times as large as the sample.interest is at least 10 times as large as the sample.

x

Page 22: SECTION 12.1

Step 3Step 3A curve should be drawn, labeled, and shaded.A curve should be drawn, labeled, and shaded.You can use the formula to calculate your t test You can use the formula to calculate your t test

statistic for this problemstatistic for this problem In this case t = -2.00In this case t = -2.00

Mark this on your sketch.Mark this on your sketch.Based on our calculations the Based on our calculations the PP-value is -value is

0.0285.0.0285. , , s=10, n=25s=10, n=25

0xt

sn

71x

Page 23: SECTION 12.1

Step 4Step 4 Since there is no predetermined level of significance Since there is no predetermined level of significance

if we are seeking to make a decision, this could be if we are seeking to make a decision, this could be argued either way. If exercisers are no different, we argued either way. If exercisers are no different, we would get results this small or smaller about 2.85% of would get results this small or smaller about 2.85% of the time by chance.the time by chance.

This result is significant at the 5% level, but is not This result is significant at the 5% level, but is not signficant at the 1% level. signficant at the 1% level.

We would likely reject HWe would likely reject H0.0. There is not much chance of obtaining a sample like There is not much chance of obtaining a sample like

we did if there is no difference, so we would reject the we did if there is no difference, so we would reject the idea that there is no difference and conclude that the idea that there is no difference and conclude that the mean diastolic blood pressure of American women mean diastolic blood pressure of American women aged 18-44 that exercise regularly is probably less aged 18-44 that exercise regularly is probably less than 75 mL Hg.than 75 mL Hg.

Page 24: SECTION 12.1

DUALITYDUALITY A level α two-sided significance test rejects a α two-sided significance test rejects a

hypothesis hypothesis HH00 : : = = 00 exactly when exactly when 00 falls outside falls outside a level 1- a level 1- α confidence interval for α confidence interval for ..

This relationship is EXACT for a TWO-SIDED This relationship is EXACT for a TWO-SIDED hypothesis test FOR A MEAN, but IS NOT EXACT hypothesis test FOR A MEAN, but IS NOT EXACT FOR tests involving PROPORTIONS.FOR tests involving PROPORTIONS.

Essentially, if the parameter value given in the null Essentially, if the parameter value given in the null hypothesis falls inside the confidence interval, then hypothesis falls inside the confidence interval, then that value is plausible. If the parameter value lands that value is plausible. If the parameter value lands outside the confidence interval, then we have good outside the confidence interval, then we have good reason to doubt reason to doubt HH0.0.

Page 25: SECTION 12.1

Matched PairsMatched Pairs(Paired t Tests)(Paired t Tests)

To compare the responses to the two To compare the responses to the two treatments in a matched pairs design, treatments in a matched pairs design, apply the one sample t procedures to the apply the one sample t procedures to the observed differencesobserved differencesMore commonly used than single-sample More commonly used than single-sample

studiesstudiesUse calculatorUse calculator

Page 26: SECTION 12.1

Example: Lean vs. ObeseExample: Lean vs. Obese Some studies have shown that lean and obese people spend Some studies have shown that lean and obese people spend

their time differently. Obese people spend fewer minutes per their time differently. Obese people spend fewer minutes per day (on average) standing and walking than do lean people day (on average) standing and walking than do lean people who are similar in age, overall health, and occupation. Is this who are similar in age, overall health, and occupation. Is this difference biological, so that it might help explain why some difference biological, so that it might help explain why some people become obese? Or is it a response to obesity—people people become obese? Or is it a response to obesity—people become less active when they gain weight?become less active when they gain weight?

A small pilot study looked at this issue. The subjects were 7 A small pilot study looked at this issue. The subjects were 7 mildly obese people who were healthy and did not follow an mildly obese people who were healthy and did not follow an exercise program. The subjects agreed to participate in a exercise program. The subjects agreed to participate in a weight-loss program for eight weeks, during which they lost and weight-loss program for eight weeks, during which they lost and average of 8 kilograms (17.6 pounds). Both before and after average of 8 kilograms (17.6 pounds). Both before and after weight loss, each subject wore monitors that recorded every weight loss, each subject wore monitors that recorded every movement for 10 days. The table on the next slide shows the movement for 10 days. The table on the next slide shows the minutes per day spent standing and walking. The response minutes per day spent standing and walking. The response variable for this study is the difference in minutes after weight variable for this study is the difference in minutes after weight loss minus minutes before weight loss. The differences appear loss minus minutes before weight loss. The differences appear in the final column of the table.in the final column of the table.

Page 27: SECTION 12.1

Time standing and walking before Time standing and walking before and after weight lossand after weight loss

Minutes per DayMinutes per Day

SubjectSubject BeforeBefore AfterAfter DifferenceDifference

11 293293 264264 -29-29

22 330330 335335 55

33 353353 387387 3434

44 354354 307307 -47-47

55 400400 387387 -13-13

66 454454 358358 -96-96

77 552552 549549 -3-3

Page 28: SECTION 12.1

Step 1—ParameterStep 1—Parameter Researcher’s question: Do mildly obese people Researcher’s question: Do mildly obese people

increase the time they spend standing and walking increase the time they spend standing and walking when they lose weight?when they lose weight?

The parameter of interest is the mean difference The parameter of interest is the mean difference (after-before) (after-before) in activity time in the entire in activity time in the entire population of such mildly obese people. The null population of such mildly obese people. The null hypothesis is “no change.” That is, the mean hypothesis is “no change.” That is, the mean difference in the entire population of mildly obese difference in the entire population of mildly obese people who lose weight is zero. The alternative people who lose weight is zero. The alternative hypothesis is that these people will increase their hypothesis is that these people will increase their activity after weight loss and therefore have a activity after weight loss and therefore have a positive difference.positive difference.

HHoo: : = 0 = 0 HHaa: : > 0 > 0

Page 29: SECTION 12.1

Step 2—ConditionsStep 2—ConditionsRandom—The 7 subjects volunteered. We Random—The 7 subjects volunteered. We

must be willing to assume that they are a must be willing to assume that they are a random sample from all people who meet random sample from all people who meet requirements for the study (mildly obese, requirements for the study (mildly obese, healthy, sedentary jobs, no exercise program, healthy, sedentary jobs, no exercise program, etc.). Human subjects are almost never etc.). Human subjects are almost never actually chosen at random from the population actually chosen at random from the population of interest, so this study is typical. We rely on of interest, so this study is typical. We rely on researchers not to bias their study by their way researchers not to bias their study by their way of choosing subjects.of choosing subjects.

Page 30: SECTION 12.1

Step 2 Continued—ConditionsStep 2 Continued—Conditions Normality—The difference -96 for Subject 6 may be a Normality—The difference -96 for Subject 6 may be a

low outlier (although it passes our standard 1.5*IQR low outlier (although it passes our standard 1.5*IQR rule). Because the observations are widely spread, it rule). Because the observations are widely spread, it is hard to judge normality from just 7 observations. is hard to judge normality from just 7 observations. The Normal probability plot suggests that these data The Normal probability plot suggests that these data could come from a Normal population.could come from a Normal population.

Independence—The differences in standing and Independence—The differences in standing and walking time for these 7 subjects should be walking time for these 7 subjects should be independent. Also, there are probably at least 70 independent. Also, there are probably at least 70 people that fall into this population allowing us to people that fall into this population allowing us to assume independence.assume independence.NOTE: The before and after measurements for each NOTE: The before and after measurements for each subject are NOT independent, which is why we use a subject are NOT independent, which is why we use a paired T-test.paired T-test.

Page 31: SECTION 12.1

Step 3—CalculationsStep 3—CalculationsWe are performing a matched pairs t-Test.We are performing a matched pairs t-Test. t = -1.3492t = -1.3492

df = n-1 = 7-1 = 6df = n-1 = 7-1 = 6PP-value = 0.8870-value = 0.8870 = -21.2857= -21.2857s = 41.7401s = 41.7401n = 7n = 7

0xt

sn

x

Don’t forget to draw your curve. Remember, this is no longer a Normal curve. Instead, we have a curve for the t-distribution. For drawing this, look at your calculator and remember it is nearly the Normal curve.

Page 32: SECTION 12.1

Step 4—InterpretationStep 4—InterpretationWith a With a PP-value this high, we would fail to reject -value this high, we would fail to reject

HH00 at any reasonable significance level. The at any reasonable significance level. The mean difference in activity time in the mean difference in activity time in the population of mildly obese people could very population of mildly obese people could very well be 0. It seems that having mildly obese well be 0. It seems that having mildly obese people lose weight may not increase their people lose weight may not increase their activity time.activity time.

NOTE: This is an unusual case where the NOTE: This is an unusual case where the value from our sample is in the opposite value from our sample is in the opposite direction from our alternative hypothesis.direction from our alternative hypothesis.

Page 33: SECTION 12.1

COMPUTER OUTPUTCOMPUTER OUTPUT Unfortunately, we rarely (or never) get the chance to Unfortunately, we rarely (or never) get the chance to

use a computer to analyze data. However, you are use a computer to analyze data. However, you are expected to be able to read computer output for the expected to be able to read computer output for the purposes of this course as well as for the AP Exam in purposes of this course as well as for the AP Exam in May.May.

The book provides several examples for you to use in The book provides several examples for you to use in your efforts to understand computer output.your efforts to understand computer output.

We will occasionally see additional examples in class.We will occasionally see additional examples in class. Most computer output is similar, so make sure you Most computer output is similar, so make sure you

know what you are looking for.know what you are looking for. Most computer output also has many numbers that Most computer output also has many numbers that

you will not use, so make sure you know which you will not use, so make sure you know which numbers matter and which ones do not.numbers matter and which ones do not.

Page 34: SECTION 12.1

Another ExampleAnother Example A medical researcher wishes to investigate the A medical researcher wishes to investigate the

effectiveness of exercise versus diet in losing weight. effectiveness of exercise versus diet in losing weight. Two groups of 25 overweight adult subjects are used, Two groups of 25 overweight adult subjects are used, with a subject in each group matched to a similar with a subject in each group matched to a similar subject in the other group on the basis of a number of subject in the other group on the basis of a number of physiological variables. One of the groups is placed physiological variables. One of the groups is placed on a regular program of vigorous exercise but with no on a regular program of vigorous exercise but with no restriction on diet, and the other on a strict diet but restriction on diet, and the other on a strict diet but with no requirement to exercise. The weight losses with no requirement to exercise. The weight losses after 20 weeks are determined for each subject, and after 20 weeks are determined for each subject, and the differences between matched pairs of subjects the differences between matched pairs of subjects (weight loss of subject in exercise group – weight loss (weight loss of subject in exercise group – weight loss of matched subject in diet group) is computed. The of matched subject in diet group) is computed. The mean of these differences in weight loss is found to be mean of these differences in weight loss is found to be -2 lb. with standard deviation s = 4 lb.-2 lb. with standard deviation s = 4 lb.

Is this evidence of a significant difference in mean Is this evidence of a significant difference in mean weight loss for the two methods? weight loss for the two methods?

Page 35: SECTION 12.1

Step 1—ParameterStep 1—ParameterLet be the mean difference in weight loss Let be the mean difference in weight loss

(exercise – diet) where the difference is for (exercise – diet) where the difference is for each pair of subjects.each pair of subjects.

The null hypothesis is that there is no difference The null hypothesis is that there is no difference in weight loss between the two methods.in weight loss between the two methods.

The alternative hypothesis is that there is a The alternative hypothesis is that there is a difference in weight loss between the two difference in weight loss between the two methods.methods.

HH00: : =0=0

HHaa: : ≠0≠0

Page 36: SECTION 12.1

Step 2—ConditionsStep 2—Conditions Random—We are not told how the subjects were chosen, so we Random—We are not told how the subjects were chosen, so we

must assume they are representative of the desired population if must assume they are representative of the desired population if we want to extend our findings to that larger population.we want to extend our findings to that larger population.

Normality—Recall, due to working with the t-distribution, when Normality—Recall, due to working with the t-distribution, when the sample size is sufficiently large, we become unconcerned the sample size is sufficiently large, we become unconcerned with the Normality of the population distribution. In this case, the with the Normality of the population distribution. In this case, the sample size is large enough to overcome some skewness, but sample size is large enough to overcome some skewness, but we would be more comfortable if we could safely assume we would be more comfortable if we could safely assume Normality in the population distribution. Of course, outliers Normality in the population distribution. Of course, outliers would damage our results.would damage our results.

Independence—We must be willing to view these 25 differences Independence—We must be willing to view these 25 differences as independent measurements or assume that there are at least as independent measurements or assume that there are at least 250 differences in the population.250 differences in the population.

Page 37: SECTION 12.1

Step 3—CalculationsStep 3—Calculations t = -2.5t = -2.5

df = 24df = 24 = -2= -2 s = 4s = 4 n = 25n = 25 PP-value = 0.0197-value = 0.0197

0xt

sn

xDon’t forget to draw your curve. Remember, this is no longer a Normal curve. Instead, we have a curve for the t-distribution. For drawing this, look at your calculator and remember it is nearly the Normal curve.

Page 38: SECTION 12.1

Step 4—InterpretationStep 4—InterpretationAssuming all conditions are satisfied:Assuming all conditions are satisfied:Because the P-value is small, we can reject HBecause the P-value is small, we can reject H0.0.

Essentially, if there is truly no difference Essentially, if there is truly no difference between the two methods, we would only get between the two methods, we would only get differences in weight loss this extreme about differences in weight loss this extreme about 1.97% of the time by chance alone. Since this 1.97% of the time by chance alone. Since this is so unlikely, we can reject the null hypothesis.is so unlikely, we can reject the null hypothesis.

Based on this evidence, there appears to be a Based on this evidence, there appears to be a difference in the average weight loss between difference in the average weight loss between the two methods.the two methods.

Page 39: SECTION 12.1

Follow Up Question to ExampleFollow Up Question to ExampleCan we conclude that a significant difference Can we conclude that a significant difference

in weight loss for the two methods is CAUSED in weight loss for the two methods is CAUSED by the specific treatment administered (diet or by the specific treatment administered (diet or exercise)? Justify your answer.exercise)? Justify your answer.

Assuming the subjects are randomly assigned Assuming the subjects are randomly assigned to each of the weight loss groups, then cause to each of the weight loss groups, then cause and effect conclusions can be drawn from this and effect conclusions can be drawn from this matched pairs experiment. For example, once matched pairs experiment. For example, once the pairings are made, the toss of a coin (or the pairings are made, the toss of a coin (or other random event) should determine which other random event) should determine which subject of each pair goes on which program.subject of each pair goes on which program.