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SECTION 12.1 Tests About a Population Mean

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Whats the difference between what is addressed in Section 11.2 (we skipped) and what we are beginning in Section 12.1? In reality, the standard deviation σ of the population is unknown, so the procedures from last chapter are not useful. However, the understanding of the logic of the procedures will continue to be of use. In reality, the standard deviation σ of the population is unknown, so the procedures from last chapter are not useful. However, the understanding of the logic of the procedures will continue to be of use. In order to be more realistic, σ is estimated from the data collected using s In order to be more realistic, σ is estimated from the data collected using s

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Overview of a Significance Test A test of significance is intended to assess the evidence provided by data against a null hypothesis H 0 in favor of an alternate hypothesis H a. A test of significance is intended to assess the evidence provided by data against a null hypothesis H 0 in favor of an alternate hypothesis H a. The statement being tested in a test of significance is called the null hypothesis. Usually the null hypothesis is a statement of no effect or no difference. The statement being tested in a test of significance is called the null hypothesis. Usually the null hypothesis is a statement of no effect or no difference. A one-sided alternate hypothesis exists when we are interested only in deviations from the null hypothesis in one direction A one-sided alternate hypothesis exists when we are interested only in deviations from the null hypothesis in one direction H 0 : =0 H a : >0 (or 0 (or <0) If the problem does not specify the direction of the difference, the alternate hypothesis is two-sided If the problem does not specify the direction of the difference, the alternate hypothesis is two-sided H 0 : =0 H a :0

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HYPOTHESES NOTE: Hypotheses ALWAYS refer to a population parameter, not a sample statistic. NOTE: Hypotheses ALWAYS refer to a population parameter, not a sample statistic. The alternative hypothesis should express the hopes or suspicions we have BEFORE we see the data. Dont cheat by looking at the data first. The alternative hypothesis should express the hopes or suspicions we have BEFORE we see the data. Dont cheat by looking at the data first.

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CONDITIONS These should be VERY FAMILIAR to you by now. These should be VERY FAMILIAR to you by now. Random Random Data is from an SRS or from a randomized experiment Data is from an SRS or from a randomized experiment Normal Normal For meanspopulation distribution is Normal or you have a large sample size (n30) to ensure a Normal sampling distribution for the sample mean For meanspopulation distribution is Normal or you have a large sample size (n30) to ensure a Normal sampling distribution for the sample mean For proportionsnp10 and n(1-p)10 (meaning the sample is large enough to ensure a Normal sampling distribution for )more details in next Section 12.2 For proportionsnp10 and n(1-p)10 (meaning the sample is large enough to ensure a Normal sampling distribution for )more details in next Section 12.2 Independent Independent 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. dev. okay. 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. dev. okay.

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CAUTION Be sure to check that the conditions for running a significance test for the population mean are satisfied before you perform any calculations. Be sure to check that the conditions for running a significance test for the population mean are satisfied before you perform any calculations.

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t Statistic note: this is the same as what we learned in Chapter 10 The statistic does not have a normal distribution Degrees of freedom: n-1 Degrees of freedom: n-1 Differs from a z statistic because σ is not used Differs from a z statistic because σ is not used t statistic says how far x is from its mean μ in standard deviation units t statistic says how far x is from its mean μ in standard deviation units We are now using the body of the table we used in the last chapter. We are now using the body of the table we used in the last chapter.

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ROBUSTNESS ROBUST: Confidence levels or P-Values do not change when some of the assumptions are violated ROBUST: Confidence levels or P-Values do not change when some of the assumptions are violated Fortunately for us, the t-procedures are robust in certain situations. Fortunately for us, the t-procedures are robust in certain situations. Therefore... Therefore...

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This is when we use the t-procedures Its more important for the data to be Its more important for the data to be an SRS from a population than the population has a normal distribution If n is less than 15, the data must be normal to use t-procedures If n is less than 15, the data must be normal to use t-procedures If n is at least 15, the t-procedures can be used except if there are outliers or strong skewness If n is at least 15, the t-procedures can be used except if there are outliers or strong skewness If n40, t-procedures can be used even in the If n40, t-procedures can be used even in the presence of strong skewness

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Density Curves for t Distributions Bell-shaped and symmetric Bell-shaped and symmetric Greater spread than a normal curve Greater spread than a normal curve As degrees of freedom (or sample size) increases, the t density curves appear more like a normal curve As degrees of freedom (or sample size) increases, the t density curves appear more like a normal curve

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Tip on Interpreting P-values On page 746, at the bottom of the box The One- Sample t Test, it is stated, These P-values are exact if the population distribution is Normal and are approximately correct for large n in other cases. Example 12.2 (same page) then uses a small sample (n=10) with no guarantee that the distribution is Normal and where the Normal probability plot is a bit iffy. Thus, this is a case in which we choose to use t where we assume that the population distribution is approximately Normal because we dont have clear evidence that it isnt. Therefore, the P-values are approximately correct. On page 746, at the bottom of the box The One- Sample t Test, it is stated, These P-values are exact if the population distribution is Normal and are approximately correct for large n in other cases. Example 12.2 (same page) then uses a small sample (n=10) with no guarantee that the distribution is Normal and where the Normal probability plot is a bit iffy. Thus, this is a case in which we choose to use t where we assume that the population distribution is approximately Normal because we dont have clear evidence that it isnt. Therefore, the P-values are approximately correct.

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INFERENCE TOOLBOX (p 705) 1PARAMETERIdentify the population of interest and the parameter you want to draw a conclusion about. STATE YOUR HYPOTHESES! 1PARAMETERIdentify the population of interest and the parameter you want to draw a conclusion about. STATE YOUR HYPOTHESES! 2CONDITIONSChoose the appropriate inference procedure. VERIFY conditions (Random, Normal, Independent) before using it. 2CONDITIONSChoose the appropriate inference procedure. VERIFY conditions (Random, Normal, Independent) before using it. 3CALCULATIONSIf the conditions are met, carry out the inference procedure. 3CALCULATIONSIf the conditions are met, carry out the inference procedure. 4INTERPRETATIONInterpret your results in the context of the problem. CONCLUSION, CONNECTION, CONTEXT(meaning that our conclusion about the parameter connects to our work in part 3 and includes appropriate context) 4INTERPRETATIONInterpret your results in the context of the problem. CONCLUSION, CONNECTION, CONTEXT(meaning that our conclusion about the parameter connects to our work in part 3 and includes appropriate context) Steps for completing a SIGNIFICANCE TEST: DO YOU REMEMBER WHAT THE STEPS ARE???

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Step 1PARAMETER Read through the problem and determine what we hope to show through our test. Read through the problem and determine what we hope to show through our test. Our null hypothesis is that no change has occurred or that no difference is evident. Our null hypothesis is that no change has occurred or that no difference is evident. Our alternative hypothesis can be either one or two sided. Our alternative hypothesis can be either one or two sided. Be certain to use appropriate symbols and also write them out in words. Be certain to use appropriate symbols and also write them out in words.

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Step 2CONDITIONS Based on the given information, determine which test should be used. Name the procedure. Based on the given information, determine which test should be used. Name the procedure. State the conditions. State the conditions. Verify (through discussion) whether the conditions have been met. For any assumptions that seem unsafe to verify as met, explain why. Dont forget, with the t distribution, there is more forgiveness due to the robustness of the t procedures Verify (through discussion) whether the conditions have been met. For any assumptions that seem unsafe to verify as met, explain why. Dont forget, with the t distribution, there is more forgiveness due to the robustness of the t procedures Remember, if data is given, graph it to help facilitate this discussion Remember, if data is given, graph it to help facilitate this discussion For each procedure there are several things that we are assuming are true that allow these procedures to produce meaningful results. For each procedure there are several things that we are assuming are true that allow these procedures to produce meaningful results.

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Step 3CALCULATIONS First write out the formula for the test statistic, report its value, mark the value on the curve. First write out the formula for the test statistic, report its value, mark the value on the curve. Sketch the density curve as clearly as possible out to three standard deviations on each side. Sketch the density curve as clearly as possible out to three standard deviations on each side. Mark the null hypothesis and sample statistic clearly on the curve. Mark the null hypothesis and sample statistic clearly on the curve. Calculate and report the P-value Calculate and report the P-value Shade the appropriate region of the curve. Shade the appropriate region of the curve. Report other values of importance (standard deviation, df, critical value, etc.) Report other values of importance (standard deviation, df, critical value, etc.)

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Step 4INTERPRETATION There are really two parts to this step: decision & conclusion. TWO UNIQUE SENTENCES. There are really two parts to this step: decision & conclusion. TWO UNIQUE SENTENCES. Based on the P-value, make a decision. Will you reject H 0 or fail to reject H 0. Based on the P-value, make a decision. Will you reject H 0 or fail to reject H 0. If there is a predetermined significance level, then make reference to this as part of your decision. If not, interpret the P-value appropriately. If there is a predetermined significance level, then make reference to this as part of your decision. If not, interpret the P-value appropriately. Now that you have made a decision, state a conclusion IN THE CONTEXT of the problem. Now that you have made a decision, state a conclusion IN THE CONTEXT of the problem. This does not need to, and probably should not, have statistical terminology involved. DO NOT use the word prove in this statement. This does not need to, and probably should not, have statistical terminology involved. DO NOT use the word prove in this statement.

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The Steps for a ONE SAMPLE t-TEST Same ApproachSlightly Different Look 1. State the hypotheses and name test H o : = 0 H a :,, or 0 2. State and verify your assumptions 3. Calculate the P-value and other important values - Done in calculator or… - Book: - 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 falls in 4. State Conclusions (Both statistically and contextually) - The smaller the P-value, the greater the evidence is to reject H o STATE PLAN DO CONCLUDE

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Summarizing the STEPS of Inference State the null and alternative hypotheses in context State the null and alternative hypotheses in context Identify the inference procedure to be used and justify the conditions for its use Identify the inference procedure to be used and justify the conditions for its use Perform statistical mechanics Perform statistical mechanics State the conclusion in the context of the problem with a clear linkage to the mechanics that imply that conclusion State the conclusion in the context of the problem with a clear linkage to the mechanics that imply that conclusion

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Example 1-sided t-Test The diastolic blood pressure for American women aged has approximately the Normal distribution with mean =75 milliliters of mercury (mL Hg) and standard deviation s=10 mL Hg. We suspect that regular exercise will lower blood pressure. A random sample of 25 women who jog at least five miles a week gives sample mean blood pressure =71 mL Hg. Is this good evidence that the mean diastolic blood pressure for the population of regular exercisers is lower than 75 mL Hg? The diastolic blood pressure for American women aged has approximately the Normal distribution with mean =75 milliliters of mercury (mL Hg) and standard deviation s=10 mL Hg. We suspect that regular exercise will lower blood pressure. A random sample of 25 women who jog at least five miles a week gives sample mean blood pressure =71 mL Hg. Is this good evidence that the mean diastolic blood pressure for the population of regular exercisers is lower than 75 mL Hg?

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Step 1 The parameter of interest is the mean diastolic blood pressure. The parameter of interest is the mean diastolic blood pressure. Our null hypothesis is that the blood pressure is no different for those that exercise. Our null hypothesis is that the blood pressure is no different for those that exercise. Our alternative hypothesis is one-sided because we suspect that exercisers have lower blood pressure. Our alternative hypothesis is one-sided because we suspect that exercisers have lower blood pressure. H 0 : = 75 mL H 0 : = 75 mL H a : < 75 mL H a : < 75 mL

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Step 2 Since we do not know the population standard deviation we will be performing a t-test of significance. Since we do not know the population standard deviation we will be performing a t-test of significance. We were told that the sample is random, but we do not know if it is an SRS from the population of interest. This may limit our ability to generalize. We were told that the sample is random, but we do not know if it is an SRS from the population of interest. This may limit our ability to generalize. Since the population distribution is approximately Normal, we know that the sampling distribution of will also be approximately Normal. So we are safe using the t procedures. Since the population distribution is approximately Normal, we know that the sampling distribution of will also be approximately Normal. So we are safe using the t procedures. The blood pressure measurements for the 25 joggers should be independent. Note that the population of interest is at least 10 times as large as the sample. The blood pressure measurements for the 25 joggers should be independent. Note that the population of interest is at least 10 times as large as the sample.

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Step 3 A 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 statistic for this problem You can use the formula to calculate your t test statistic for this problem In this case t = In this case t = Mark this on your sketch. Mark this on your sketch. Based on our calculations the P-value is Based on our calculations the P-value is , s=10, n=25, s=10, n=25

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Step 4 Since there is no predetermined level of significance if we are seeking to make a decision, this could be argued either way. If exercisers are no different, we would get results this small or smaller about 2.85% of the time by chance. Since there is no predetermined level of significance if we are seeking to make a decision, this could be argued either way. If exercisers are no different, we would get results this small or smaller about 2.85% of the time by chance. This result is significant at the 5% level, but is not signficant at the 1% level. This result is significant at the 5% level, but is not signficant at the 1% level. We would likely reject H 0. We would likely reject H 0. There is not much chance of obtaining a sample like we did if there is no difference, so we would reject the idea that there is no difference and conclude that the mean diastolic blood pressure of American women aged that exercise regularly is probably less than 75 mL Hg. There is not much chance of obtaining a sample like we did if there is no difference, so we would reject the idea that there is no difference and conclude that the mean diastolic blood pressure of American women aged that exercise regularly is probably less than 75 mL Hg.

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DUALITY α two-sided significance test rejects a hypothesis H 0 : = 0 exactly when 0 falls outside a level 1- α confidence interval for. A level α two-sided significance test rejects a hypothesis H 0 : = 0 exactly when 0 falls outside a level 1- α confidence interval for. This relationship is EXACT for a TWO-SIDED hypothesis test FOR A MEAN, but IS NOT EXACT FOR tests involving PROPORTIONS. This relationship is EXACT for a TWO-SIDED hypothesis test FOR A MEAN, but IS NOT EXACT FOR tests involving PROPORTIONS. Essentially, if the parameter value given in the null hypothesis falls inside the confidence interval, then that value is plausible. If the parameter value lands outside the confidence interval, then we have good reason to doubt H 0. Essentially, if the parameter value given in the null hypothesis falls inside the confidence interval, then that value is plausible. If the parameter value lands outside the confidence interval, then we have good reason to doubt H 0.

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Matched Pairs (Paired t Tests) To compare the responses to the two treatments in a matched pairs design, apply the one sample t procedures to the observed differences To compare the responses to the two treatments in a matched pairs design, apply the one sample t procedures to the observed differences More commonly used than single-sample studies More commonly used than single-sample studies Use calculator Use calculator

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Example: Lean vs. Obese Some studies have shown that lean and obese people spend their time differently. Obese people spend fewer minutes per day (on average) standing and walking than do lean people who are similar in age, overall health, and occupation. Is this difference biological, so that it might help explain why some people become obese? Or is it a response to obesitypeople become less active when they gain weight? Some studies have shown that lean and obese people spend their time differently. Obese people spend fewer minutes per day (on average) standing and walking than do lean people who are similar in age, overall health, and occupation. Is this difference biological, so that it might help explain why some people become obese? Or is it a response to obesitypeople become less active when they gain weight? A small pilot study looked at this issue. The subjects were 7 mildly obese people who were healthy and did not follow an exercise program. The subjects agreed to participate in a weight-loss program for eight weeks, during which they lost and average of 8 kilograms (17.6 pounds). Both before and after weight loss, each subject wore monitors that recorded every movement for 10 days. The table on the next slide shows the minutes per day spent standing and walking. The response variable for this study is the difference in minutes after weight loss minus minutes before weight loss. The differences appear in the final column of the table. A small pilot study looked at this issue. The subjects were 7 mildly obese people who were healthy and did not follow an exercise program. The subjects agreed to participate in a weight-loss program for eight weeks, during which they lost and average of 8 kilograms (17.6 pounds). Both before and after weight loss, each subject wore monitors that recorded every movement for 10 days. The table on the next slide shows the minutes per day spent standing and walking. The response variable for this study is the difference in minutes after weight loss minus minutes before weight loss. The differences appear in the final column of the table.

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Time standing and walking before and after weight loss Minutes per Day SubjectBeforeAfterDifference

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Step 1Parameter Researchers question: Do mildly obese people increase the time they spend standing and walking when they lose weight? Researchers question: Do mildly obese people increase the time they spend standing and walking when they lose weight? The parameter of interest is the mean difference (after-before) in activity time in the entire population of such mildly obese people. The null hypothesis is no change. That is, the mean difference in the entire population of mildly obese people who lose weight is zero. The alternative hypothesis is that these people will increase their activity after weight loss and therefore have a positive difference. The parameter of interest is the mean difference (after-before) in activity time in the entire population of such mildly obese people. The null hypothesis is no change. That is, the mean difference in the entire population of mildly obese people who lose weight is zero. The alternative hypothesis is that these people will increase their activity after weight loss and therefore have a positive difference. H o : = 0 H o : = 0 H a : > 0 H a : > 0

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Step 2Conditions RandomThe 7 subjects volunteered. We must be willing to assume that they are a random sample from all people who meet requirements for the study (mildly obese, healthy, sedentary jobs, no exercise program, etc.). Human subjects are almost never actually chosen at random from the population of interest, so this study is typical. We rely on researchers not to bias their study by their way of choosing subjects. RandomThe 7 subjects volunteered. We must be willing to assume that they are a random sample from all people who meet requirements for the study (mildly obese, healthy, sedentary jobs, no exercise program, etc.). Human subjects are almost never actually chosen at random from the population of interest, so this study is typical. We rely on researchers not to bias their study by their way of choosing subjects.

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Step 2 ContinuedConditions NormalityThe difference -96 for Subject 6 may be a low outlier (although it passes our standard 1.5*IQR rule). Because the observations are widely spread, it is hard to judge normality from just 7 observations. The Normal probability plot suggests that these data could come from a Normal population. NormalityThe difference -96 for Subject 6 may be a low outlier (although it passes our standard 1.5*IQR rule). Because the observations are widely spread, it is hard to judge normality from just 7 observations. The Normal probability plot suggests that these data could come from a Normal population. IndependenceThe differences in standing and walking time for these 7 subjects should be independent. Also, there are probably at least 70 people that fall into this population allowing us to assume independence. IndependenceThe differences in standing and walking time for these 7 subjects should be independent. Also, there are probably at least 70 people that fall into this population allowing us to assume independence. NOTE: The before and after measurements for each subject are NOT independent, which is why we use a paired T-test.

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Step 3Calculations We are performing a matched pairs t-Test. We are performing a matched pairs t-Test. t = t = df = n-1 = 7-1 = 6 df = n-1 = 7-1 = 6 P-value = P-value = = = s = s = n = 7 n = 7 Dont 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.

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Step 4Interpretation With a P-value this high, we would fail to reject H 0 at any reasonable significance level. The mean difference in activity time in the population of mildly obese people could very well be 0. It seems that having mildly obese people lose weight may not increase their activity time. With a P-value this high, we would fail to reject H 0 at any reasonable significance level. The mean difference in activity time in the population of mildly obese people could very well be 0. It seems that having mildly obese people lose weight may not increase their activity time. NOTE: This is an unusual case where the value from our sample is in the opposite direction from our alternative hypothesis. NOTE: This is an unusual case where the value from our sample is in the opposite direction from our alternative hypothesis.

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COMPUTER OUTPUT Unfortunately, we rarely (or never) get the chance to use a computer to analyze data. However, you are expected to be able to read computer output for the purposes of this course as well as for the AP Exam in May. Unfortunately, we rarely (or never) get the chance to use a computer to analyze data. However, you are expected to be able to read computer output for the purposes of this course as well as for the AP Exam in May. The book provides several examples for you to use in your efforts to understand computer output. The book provides several examples for you to use in 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 know what you are looking for. Most computer output is similar, so make sure you know what you are looking for. Most computer output also has many numbers that you will not use, so make sure you know which numbers matter and which ones do not. Most computer output also has many numbers that you will not use, so make sure you know which numbers matter and which ones do not.

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

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Step 1Parameter Let be the mean difference in weight loss (exercise – diet) where the difference is for each pair of subjects. Let be the mean difference in weight loss (exercise – diet) where the difference is for each pair of subjects. The null hypothesis is that there is no difference in weight loss between the two methods. The null hypothesis is that there is no difference in weight loss between the two methods. The alternative hypothesis is that there is a difference in weight loss between the two methods. The alternative hypothesis is that there is a difference in weight loss between the two methods. H 0 : =0 H a : 0

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Step 2Conditions RandomWe are not told how the subjects were chosen, so we must assume they are representative of the desired population if we want to extend our findings to that larger population. RandomWe are not told how the subjects were chosen, so we must assume they are representative of the desired population if we want to extend our findings to that larger population. NormalityRecall, due to working with the t-distribution, when the sample size is sufficiently large, we become unconcerned with the Normality of the population distribution. In this case, the sample size is large enough to overcome some skewness, but we would be more comfortable if we could safely assume Normality in the population distribution. Of course, outliers would damage our results. NormalityRecall, due to working with the t-distribution, when the sample size is sufficiently large, we become unconcerned with the Normality of the population distribution. In this case, the sample size is large enough to overcome some skewness, but we would be more comfortable if we could safely assume Normality in the population distribution. Of course, outliers would damage our results. IndependenceWe must be willing to view these 25 differences as independent measurements or assume that there are at least 250 differences in the population. IndependenceWe must be willing to view these 25 differences as independent measurements or assume that there are at least 250 differences in the population.

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Step 3Calculations t = -2.5 t = -2.5 df = 24 df = 24 = -2 = -2 s = 4 s = 4 n = 25 n = 25 P-value = P-value = Dont 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.

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Step 4Interpretation Assuming all conditions are satisfied: Assuming all conditions are satisfied: Because the P-value is small, we can reject H 0. Because the P-value is small, we can reject H 0. Essentially, if there is truly no difference between the two methods, we would only get differences in weight loss this extreme about 1.97% of the time by chance alone. Since this is so unlikely, we can reject the null hypothesis. Essentially, if there is truly no difference between the two methods, we would only get differences in weight loss this extreme about 1.97% of the time by chance alone. Since this is so unlikely, we can reject the null hypothesis. Based on this evidence, there appears to be a difference in the average weight loss between the two methods. Based on this evidence, there appears to be a difference in the average weight loss between the two methods.

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Follow Up Question to Example Can we conclude that a significant difference in weight loss for the two methods is CAUSED by the specific treatment administered (diet or exercise)? Justify your answer. Can we conclude that a significant difference in weight loss for the two methods is CAUSED by the specific treatment administered (diet or exercise)? Justify your answer. Assuming the subjects are randomly assigned to each of the weight loss groups, then cause and effect conclusions can be drawn from this matched pairs experiment. For example, once the pairings are made, the toss of a coin (or other random event) should determine which subject of each pair goes on which program. Assuming the subjects are randomly assigned to each of the weight loss groups, then cause and effect conclusions can be drawn from this matched pairs experiment. For example, once the pairings are made, the toss of a coin (or other random event) should determine which subject of each pair goes on which program.

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