Sociology 6Z03 Topic 15: Statistical Inference for Means

Transcription

1 Sociology 6Z03 Topic 15: Statistical Inference for Means John Fox McMaster University Fall 2016 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Outline: Statistical Inference for Means Introduction Comparing Two Means From Independent Samples John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

2 Introduction We have learned how to perform classical statistical inference constructing confidence intervals and testing hypotheses for a population mean µ when (unrealistically) the population standard deviation σ is known. Our procedures made use of the fact that the sample mean has the approximate sampling distribution N(µ, σ/ n). Almost regardless of the population distribution of X, this approximation grows more accurate as the sample size n grows (the central limit theorem); and if the population distribution of X is normal i.e., X N(µ, σ) then the result is exact, even for small samples. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Introduction In this lecture, we will learn how to perform statistical inference for a population mean µ when the population standard deviation σ is not known. We will also learn how to construct confidence intervals and perform hypothesis tests for the difference between two means from independently sampled populations a common research situation. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

3 Assumptions The procedures that we are about to develop depend upon two key assumptions: 1 The data are a simple random sample from a much larger population. 2 The distribution of the variable in the population is a normal distribution with mean µ and standard deviation σ, both of which are unknown. If the distribution is single-peaked, roughly symmetric, and doesn t have very heavy tails (which tend to give rise to outliers), the assumption of normality isn t critical unless the sample size is very small. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 The Standard Error of the Sample Mean Although sample means are normally distributed with mean µ and standard deviation σ/ n, we cannot make direct use of this fact when σ is not known. We can, however, estimate the standard deviation of x by using the sample standard deviation s in place of the unknown σ. The resulting estimated standard deviation of x is called the standard error of x: SE(x) = s n A note on terminology: Some authors refer to SD(x) = σ/ n as the standard error of x ; then s/ n is called the estimated standard error of x. Following Moore, I will reserve the term standard error for the estimate that is, s/ n. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

4 The t Distributions When the population standard deviation σ is known, tests and confidence intervals for µ are based on the standardized sample mean z = x µ σ/ n where z N(0, 1). When σ is not known, we can calculate the analogous statistic, t = x µ s/ n but this statistic is not normally distributed. Instead, if the population distribution of X is a normal distribution, then this new statistic follows Student s t-distribution with n 1 degrees of freedom. Student was the nom-de-plume of the discoverer of the t-distribution W. S. Gosset, a statistician who worked for Guinness brewery around the turn of the 20th century. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 The t Distributions There is a different t-distribution for each number of degrees of freedom, 1, 2,.... The degrees of freedom for t come from the denominator of the sample standard deviation s, (x s = i x) 2 n 1 As the degrees of freedom grow, the t-distribution approaches the normal distribution. For small degrees of freedom, the t-distribution is more spread out than the normal distribution, reflecting the additional uncertainty that results from having to estimate σ rather than knowing it exactly. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

5 Some t Distributions Density N(0,1) t-distributions: N(0, 1) = t( ). t(10) t(2) John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Confidence Intervals and Hypothesis Tests Confidence intervals and tests based on the t-distribution are very similar to intervals and tests based on the normal distribution. Rather than using critical values from the normal distribution, however, we need to use critical values of t. These values may be found for various degrees of freedom in Table C of the text. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

6 t Confidence Intervals and Hypothesis Tests: Example Consider the following (familiar) example: An educational researcher wants to know whether a new method of teaching statistics is superior to the old method. Ten instructors who each teach two sections of an introductory statistics class are recruited into the study. Each instructor has one of his or her sections assigned at random to the new teaching method; the other section is taught by the old method. At the end of the study, the students in all sections of the course take a common exam. The average grade on the exam in each section is contained in the table on the next slide, along with the difference for each instructor between the scores for the sections taught according to the new and old methods. We previously analyzed these data without the proper tools, by simply pretending that the population standard deviation σ is the same as the sample standard deviation s. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Confidence Intervals and Hypothesis Tests: Example Instructor New Method Old Method Difference Class Mean Class Mean x i The mean difference is x = 9.40, and the standard deviation of the n = 10 differences is s = John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

7 t Confidence Interval: Example Using the t distribution to construct a 95-percent confidence interval: Degrees of freedom = n 1 = 10 1 = 9. The critical value of t with 9 degrees of freedom for the C =.95 level of confidence has the area.025 to the right; from Table C, this value is t = t* 0 t* = t with 9 d.f. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Confidence Interval: Example Thought Question TRUE or FALSE: t = is larger than the corresponding critical value of z (which is z = 1.960), and therefore t produces a narrower, more precise confidence interval than when the population standard deviation σ is known. A TRUE B FALSE C I don t know. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

8 t Confidence Interval: Example The 95-percent confidence interval is x ± t s = 9.40 ± n 10 = 9.40 ± 5.99 = 3.41 to points John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Hypothesis Test: Example Alternatively, to test the null hypothesis that the new method is no better than the old H 0 : µ = 0 against the alternative hypothesis that the new method is better we calculate the test statistic H a : µ > 0 t = x µ 0 s/ n = / 10 = If the null hypothesis is true, then this test statistic follows a t distribution with n 1 = 10 1 = 9 degrees of freedom. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

9 t Hypothesis Test: Example Most computer programs report the exact P-value for a t-test, but if we need to use the t-table we won t usually be able to find a precise P-value. Because the alternative hypothesis is directional, and specifies a positive value of µ, we find the P-value by looking in the right-hand tail of the t-distribution with 9 degrees of freedom: One-Sided P t John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Hypothesis Test: Example Thought Question The observed value of t is What is the P-value for the hypothesis test? A.005 < P <.0025 B.0025 < P <.005 C P =.001 D I don t know. One-Sided P t John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

10 t Hypothesis Test: Example P t with 9 d.f. 0 t* = prob. to the right =.005 observed t = Finding the P-value. t* = prob. to the right =.0025 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Hypothesis Test: Example, Two-Sided Test Thought Question TRUE or FALSE: If the alternative hypothesis were nondirectional, H a : µ = 0, then, in comparison with the previous one-sided test, we would double the P-value, which would be reported as t = 3.547, df = 9,.005 < P <.01, two-tail. A TRUE B FALSE C I don t know. One-Sided P Two-Sided P t John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

11 t Confidence Intervals and Hypothesis Tests: Caution A practical limitation of t-tests and t-intervals is that in small samples they depend upon the assumption that the population is normally distributed. In large samples, t-tests and t-intervals are generally quite accurate even if the population is not normal, but in large samples inferences based on the t-distribution and the normal distribution are essentially indistinguishable. We should check the assumption of normality by examining the distribution of the data, but in small samples where the assumption really counts it is hard to assess departures from normality. We should, however, be on the lookout for outliers, more than one mode, and serious skewness. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Confidence Intervals and Hypothesis Tests: Caution A stem-plot for the illustrative dataset: There s nothing obviously problematic here: The distribution is single-peaked, roughly symmetric, and has no outliers. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

12 t Confidence Intervals and Hypothesis Tests: Matched Pairs A very common use of t-intervals and t-tests for a single mean is for matched-pairs data. The illustration above is an example of matched pairs: Each instructor taught two classes, one of which was taught according to the old method, and the other according to the new method. The design of the study would be fundamentally different if the 20 classes were taught by 20 different instructors. In this case, 10 instructors could be randomly assigned to teach by the new method, 10 by the old method. This alternative study uses two independent samples rather than matched pairs. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Confidence Intervals and Hypothesis Tests: Matched Pairs Some other common examples of matched pairs: Before-After Studies: We examine the average auto accident rate x in n = 10 jurisdictions before and after the imposition of random spot checks. Note that this is an inherently weak design, because we do not control for what would have happened had the spot checks not started. A comparative experiment, in which some jurisdictions have spot checks imposed and others not, would be a better design. Sampling of Natural Pairs: We sample n = 100 heterosexual married couples and calculate the average difference x between husbands and wives incomes. This is different from sampling husbands and wives (or men and women) independently. Where they are appropriate, matched-pairs designs tend to be more powerful than independent-samples designs, because each pair serves as its own control. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

13 Comparing Two Means From Independent Samples As mentioned, comparing two means from independent samples is a very common research situation. We will proceed under the following assumptions: 1 We have two independent simple random samples from two different populations. Matched or paired samples are examples of dependent samples. Often a simple random sample of a general population is divided into two independent subsamples. For example, a general simple random sample of the adult Canadian population can be divided into independent subsamples of men and women. 2 Each population is normally distributed, but with unknown means and standard deviations. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Notation and Null Hypothesis We will use the following notation to describe the populations: Population Variable Mean Standard Deviation 1 x 1 µ 1 σ 1 2 x 2 µ 2 σ 2 Our interest is in comparing the population means µ 1 and µ 2. We can do this either by constructing a confidence interval for the difference µ 1 µ 2 or by testing the null hypothesis of equal population means, H 0 : µ 1 = µ 2 which is equivalent to the null hypothesis of no difference in population means, H 0 : µ 1 µ 2 = 0 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

14 Comparing Two Means From Independent Samples Sample Data We can lay out our sample data as follows: Population Sample Size Sample Mean Sample Std. Dev. 1 n 1 x 1 s 1 2 n 2 x 2 s 2 It is natural to use the sample difference x 1 x 2 to estimate the population difference µ 1 µ 2. Because the population standard deviations σ 1 and σ 2 are unknown, we will use the corresponding sample standard deviations s 1 and s 2 to estimate them. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Sampling Distribution of the Difference in Sample Means To perform statistical inference for the difference µ 1 µ 2 based upon x 1 x 2, we need to know the properties of the sampling distribution of x 1 x 2 : The mean of x 1 x 2 is µ 1 µ 2. The variance of x 1 x 2 is so the standard deviation of x 1 x 2 is σ 2 1 n 1 + σ2 2 n 2 SD(x 1 x 2 ) = σ 2 1 n 1 + σ2 2 n 2 If the two populations are normally distributed then so is the difference in sample means, x 1 x 2. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

15 Comparing Two Means From Independent Samples Sampling Distribution of the Difference in Sample Means Thought Question TRUE or FALSE: The difference in sample means x 1 x 2 is an unbiased estimator of the difference in population means µ 1 µ 2. A TRUE B FALSE C I don t know. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Two-Sample t-tests and t-intervals If the population standard deviations σ 1 and σ 2 were known, then we could base statistical inference on the normal distribution, because the standardized value z = (x 1 x 2 ) (µ 1 µ 2 ) σ1 2 + σ2 2 n 1 n 2 follows the standard normal distribution, N(0, 1). John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

16 Comparing Two Means From Independent Samples Two-Sample t-tests and t-intervals Instead, we use the two-sample t-statistic t = (x 1 x 2 ) (µ 1 µ 2 ) s1 2 + s2 2 n 1 n 2 Under the assumptions of simple random sampling and normal populations this statistic follows an approximate t-distribution. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Two-Sample t Confidence Interval To construct a level-c confidence interval, calculate (x 1 x 2 ) ± t s1 2 + s2 2 n 1 n 2 where t is the appropriate critical value from the t-distribution with degrees of freedom equal to the smaller of n 1 1 and n 2 1 (or estimated by a complex formula, called the Welch-Satterthwaite equation, given in the text). John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

17 Comparing Two Means From Independent Samples Two-Sample t Hypothesis Test To test the null hypothesis H 0 : µ 1 = µ 2, calculate the statistic t = x 1 x 2 s1 2 + s2 2 n 1 n 2 and refer this statistic to the t-distribution with degrees of freedom equal to the smaller of n 1 1 and n 2 1 (or estimated by the Welch-Satterthwaite formula). Notice that the numerator of this test statistic comes from (x 1 x 2 ) 0; That is, 0 is the hypothesized value of µ 1 µ 2. We perform a one-sided or two-sided test depending upon whether the alternative hypothesis is directional, or nondirectional, H a : µ 1 = µ 2. H a : µ 1 > µ 2 or H a : µ 1 < µ 2, John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Example To illustrate the two-sample t procedures, I ve employed data drawn from a survey of sociology students at McMaster, dividing the students who responded to the survey into two groups: (1) Those with grade-point averages of B or less; and (2) those with grade-point averages of B+ or better. In each group, I ve calculated the mean and standard deviation of number of hours of TV viewing per week. The results are as follows: Grade-Point Average n x s B or lower B+ or higher John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

18 Comparing Two Means From Independent Samples Example: 95 Percent Confidence Interval The smaller of n 1 1 and n 2 1 is 45 1 = 44. The critical value of t with 40 d.f. the closest value below 44 d.f. in the t table is t = The 95 percent confidence interval is therefore (x 1 x 2 ) ± t s1 2 + s2 2 n 1 n = ( ) ± = 4.57 ± 3.67 = 0.90 to 8.24 hours John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Example: Hypothesis Test To test H 0 : µ 1 = µ 2 against the one-sided alternative hypothesis H a : µ 1 > µ 2 (students with lower grades watch more TV), calculate t = x 1 x 2 s1 2 + s2 2 n 1 n 2 = = John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

19 Comparing Two Means From Independent Samples Example: Hypothesis Test Thought Question Entering the t table with 40 d.f. for our test statistic t = 2.514, what is the P-value for the test? A.025 < P <.05. B.01 < P <.02. C.005 < P <.01. D I don t know. One-Sided P Two-Sided P t John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Example: Hypothesis Test Thought Question If we specified a two-sided alternative hypothesis, what would be the P-value associated with our test statistic t = 2.514? A.005 < P <.01. B.01 < P <.02. C.0025 < P <.005. D I don t know. One-Sided P Two-Sided P t John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

20 TV Hours/Week Comparing Two Means From Independent Samples Example: Checking the Data Examining the data graphically reveals problems: The distribution of hours of TV watching within groups is somewhat positively skewed and there are outliers in both groups (with the dots representing the group means): B or lower B+ or higher Grade Point Average John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Example: Checking the Data With samples as large as 95 and 45, the t test and t interval are approximately correct even when the distributions are quite skewed (and hence non-normal). We say that the validity of the t procedures is robust with respect to departures from the assumption of normality. We might wonder, however, whether it would be better to use the group medians, rather than the group means, to summarize the data. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

21 Comparing Two Means From Independent Samples Example: Checking the Data As well, eliminating the outliers changes the results somewhat: Grade-Point Average n x s B or lower B+ or higher t = 5.19 df 42 1 = 41 P <.0005 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41