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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?.
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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? • 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
Overview of a Significance Test • A test of significance is intended to assess the evidence provided by data against a null hypothesis H0 in favor of an alternate hypothesis Ha. • 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 H0 : =0 Ha : >0 (or <0) • If the problem does not specify the direction of the difference, the alternate hypothesis is two-sided H0: =0 Ha: ≠0
HYPOTHESES • 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. Don’t “cheat” by looking at the data first.
CONDITIONS • These should be VERY FAMILIAR to you by now. • Random • Data is from an SRS or from a randomized experiment • Normal • For means—population distribution is Normal or you have a large sample size (n≥30) to ensure a Normal sampling distribution for the sample mean • 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 next Section 12.2 • 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.
CAUTION • Be sure to check that the conditions for running a significance test for the population mean are satisfied before you perform any calculations.
t Statisticnote: this is the same as what we learned in Chapter 10 The statistic does not have a normal distribution • Degrees of freedom: n-1 • 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 • We are now using the body of the table we used in the last chapter.
ROBUSTNESS • 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. • Therefore . . .
This is when we use the t-procedures • It’s 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 at least 15, the t-procedures can be used except if there are outliers or strong skewness • If n≥40, t-procedures can be used even in the presence of strong skewness
Density Curves for t Distributions • Bell-shaped and symmetric • Greater spread than a normal curve • As degrees of freedom (or sample size) increases, the t density curves appear more like a normal curve
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 don’t have clear evidence that it isn’t. Therefore, the P-values are approximately correct.
INFERENCE TOOLBOX (p 705) DO YOU REMEMBER WHAT THE STEPS ARE??? Steps for completing a SIGNIFICANCE TEST: • 1—PARAMETER—Identify the population of interest and the parameter you want to draw a conclusion about. STATE YOUR HYPOTHESES! • 2—CONDITIONS—Choose the appropriate inference procedure. VERIFY conditions (Random, Normal, Independent) before using it. • 3—CALCULATIONS—If the conditions are met, carry out the inference procedure. • 4—INTERPRETATION—Interpret 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)
Step 1—PARAMETER • 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 alternative hypothesis can be either one or two sided. • Be certain to use appropriate symbols and also write them out in words.
Step 2—CONDITIONS • Based on the given information, determine which test should be used. Name the procedure. • State the conditions. • Verify (through discussion) whether the conditions have been met. For any assumptions that seem unsafe to verify as met, explain why. Don’t 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 • For each procedure there are several things that we are assuming are true that allow these procedures to produce meaningful results.
Step 3—CALCULATIONS • 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. • Mark the null hypothesis and sample statistic clearly on the curve. • Calculate and report the P-value • Shade the appropriate region of the curve. • Report other values of importance (standard deviation, df, critical value, etc.)
Step 4—INTERPRETATION • There are really two parts to this step: decision & conclusion. TWO UNIQUE SENTENCES. • Based on the P-value, make a decision. Will you reject H0 or fail to reject H0. • 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. • This does not need to, and probably should not, have statistical terminology involved. DO NOT use the word “prove” in this statement.
The Steps for a ONE SAMPLE t-TESTSame Approach—Slightly Different Look • State the hypotheses and name test Ho: = 0 Ha: ‹, ›, or ≠ 0 • State and verify your assumptions • 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 • State Conclusions (Both statistically and contextually) - The smaller the P-value, the greater the evidence is to reject Ho STATE PLAN DO CONCLUDE
Summarizing the STEPS of Inference • State the null and alternative hypotheses in context • Identify the inference procedure to be used and justify the conditions for its use • Perform statistical mechanics • State the conclusion in the context of the problem with a clear linkage to the mechanics that imply that conclusion
Example 1-sided t-Test • The diastolic blood pressure for American women aged 18-44 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?
Step 1 • 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 alternative hypothesis is one-sided because we suspect that exercisers have lower blood pressure. • H0: = 75 mL • Ha: < 75 mL
Step 2 • 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. • 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.
Step 3 • A curve should be drawn, labeled, and shaded. • You can use the formula to calculate your t test statistic for this problem • In this case t = -2.00 • Mark this on your sketch. • Based on our calculations the P-value is 0.0285. • , s=10, n=25
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. • This result is significant at the 5% level, but is not signficant at the 1% level. • We would likely reject H0. • 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 18-44 that exercise regularly is probably less than 75 mL Hg.
DUALITY • A level α two-sided significance test rejects a hypothesis H0 : = 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. • 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 H0.
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 • More commonly used than single-sample studies • Use calculator
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 obesity—people 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.
Step 1—Parameter • Researcher’s 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. Ho: = 0 Ha: > 0
Step 2—Conditions • Random—The 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.
Step 2 Continued—Conditions • Normality—The 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. • Independence—The 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.
Step 3—Calculations • We are performing a matched pairs t-Test. • t = -1.3492 • df = n-1 = 7-1 = 6 • P-value = 0.8870 • = -21.2857 • s = 41.7401 • n = 7 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.
Step 4—Interpretation • With a P-value this high, we would fail to reject H0 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.
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. • The book provides several examples for you to use in your efforts to understand computer output. • 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 also has many numbers that you will not use, so make sure you know which numbers matter and which ones do not.
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. • Is this evidence of a significant difference in mean weight loss for the two methods?
Step 1—Parameter • 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 alternative hypothesis is that there is a difference in weight loss between the two methods. • H0: =0 • Ha: ≠0
Step 2—Conditions • Random—We 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. • Normality—Recall, 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. • Independence—We must be willing to view these 25 differences as independent measurements or assume that there are at least 250 differences in the population.
Step 3—Calculations • t = -2.5 • df = 24 • = -2 • s = 4 • n = 25 • P-value = 0.0197 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.
Step 4—Interpretation • Assuming all conditions are satisfied: • Because the P-value is small, we can reject H0. • 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.
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. • 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.
