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Week 9 Confidence Intervals and Tests About One Mean Using sampling distributions to make inferences Now we are ready to begin learning how to apply probability theory and sampling distributions in order to make inferences about populations based on samples. This week we will learn two techniques for doing so: Using a single sample to estimate a “confidence interval” in which a population mean falls. Testing a statistical hypothesis about a mean in a single population using a single sample. Confidence intervals Say we are interested in using our sample data to estimate the mean on some variable in a population. There are many examples of situations where the mean in a population is of interest: The mean GPA of the students at a university, based on a sample of students The actual proportion of registered voters who plan to vote for Obama in the next election, based on a national survey The mean earnings of African American males, based on a national survey The mean SAT score of applicants to the UW, based on a randomly drawn sample of 250 applications The average mpg of a production run of 10,000 autos of a particular make produced during 2003, based on a sample of 150 autos The average number of times residents in poor urban neighborhoods are victimized by crimes in the course of a year, based on a survey of residents in poor neighborhoods The average length of criminal sentences handed down to those convicted of white collar crimes, based on a sample of sentencing decisions Confidence intervals Remember: The mean from a particular sample is only an estimate of the population mean. If we were to draw repeated samples of the same size from the population, we would obtain multiple sample means that have a distribution, i.e., the sampling distribution of sample means. The sampling distribution of sample means is approximately normal, with: 1. x x 2 . x x N We can use these properties of the sampling distribution of sample means to determine what interval the population mean probably falls in, based on the sample mean and sample standard deviation. Confidence intervals What do we mean by “probably falls”? We can specify very precisely what mean by “probably.” This is the level of confidence we have in our interval. For example, we can determine the interval in which we can be 95% confident that the population mean falls. We call such an interval the “95% confidence interval” Confidence intervals What does “95% confidence” mean? Intuitive interpretation: we are 95% confident that x falls in that range Statistical interpretation: if we apply 95% confidence intervals consistently, then we will (over the long run) reach the correct conclusion about x in 95 out of 100 cases By adopting a 95% confidence level, we are recognizing that in about 5 out of 100 cases our conclusion about x will be incorrect – that is, it will outside of the confidence interval. Confidence intervals How do we compute a confidence interval? We begin by applying the t-distribution to answer the question: what is the range of values of t such that 5% of the observations will fall outside that range, i.e., what is ta= t.05? We look that up in Statistical Table C in the “twotailed test” column. For example, for a .05 and df > 120, ta= 1.96. For a .05 and df = 18, ta= 2.10. For a .01 and df = 18, ta= 2.88. Confidence intervals The critical value of t for a .05 and df > 120 of 1.96 tells us that 5% of the observations will fall 1.96 or more from the mean (either above or below) on the t-scale, which means that 95% of the observations will fall within 1.96 of the mean (above or below). In one sense, all 95% confidence intervals for sample sizes greater than 120 are ±1.96 on the t-scale. But what does “on the t-scale” mean? We need to convert the t-value of ±1.96 into values in the original scale of the variable in question. To do this, think of how you convert a raw score to a t-score: xx tx sx Confidence intervals Keeping in mind that the standard deviation of a sampling distribution is what we call the standard error, the ttransformation of a single-sample mean deviation is obtained by dividing it by the standard error:s s x x Thus: xx xx tx sx sx n 1 n 1 Confidence intervals If xx xx ta 1.96 , then sx sx n 1 sx x x 1.96 * n 1 and sx x x 1.96 * n 1 Confidence intervals How do we compute a confidence interval? 100 * (1 a )% Confidence interval of x x (ta )( sx ) So, we have to determine: a (level of significance, or 1-confidence, or probability of being outside the interval) x (sample mean) ta (critical value of t for a given a and degrees-offreedom) s x (standard error, estimated using the sample standard deviation) Confidence intervals 100 * (1 a )% Confidence interval of x x (ta )( sx ) Example: A random sample of 61 Sociology majors has a mean GPA of 3.2, with a standard deviation of .35. Compute and interpret the 95% confidence interval of the mean GPA for Sociology majors. 1) 2) 3) Determine that a is .05. Determine that ta is 2.00 (what are the degrees of freedom?) Plug the numbers into the formula and solve: 95 % CI of x 3.2 2.00 * (.35 / 60 ) 3.12 to 3.28 Interpret: we are 95% confident that the mean GPA for Sociology majors is between 3.12 and 3.28. Confidence intervals 100 * (1 a )% Confidence interval of x x (ta )( sx ) Another example: A random sample of 10 “Guzzler” SUV‟s produced at Factory K has a mean mpg of 5, with a standard deviation of .18. Compute and interpret the 95% confidence interval of the mean mpg of the Guzzlers manufactured at that plant. Confidence intervals 100 * (1 a )% Confidence interval of x x (ta )( sx ) Another example: A random sample of 10 “Guzzler” SUV‟s produced at Factory K has a mean mpg of 5, with a standard deviation of .18. Compute and interpret the 95% confidence interval of the mean mpg of the Guzzlers manufactured at that plant. 1) What is a? Confidence intervals 100 * (1 a )% Confidence interval of x x (ta )( sx ) Another example: A random sample of 10 “Guzzler” SUV‟s produced at Factory K has a mean mpg of 5, with a standard deviation of .18. Compute and interpret the 95% confidence interval of the mean mpg of the Guzzlers manufactured at that plant. 1) What is a? .05 Confidence intervals 100 * (1 a )% Confidence interval of x x (ta )( sx ) Another example: A random sample of 10 “Guzzler” SUV‟s produced at Factory K has a mean mpg of 5, with a standard deviation of .18. Compute and interpret the 95% confidence interval of the mean mpg of the Guzzlers manufactured at that plant. 1) 2) What is a? .05 What is ta ? (What are the degrees of freedom?) Confidence intervals 100 * (1 a )% Confidence interval of x x (ta )( sx ) Another example: A random sample of 10 “Guzzler” SUV‟s produced at Factory K has a mean mpg of 5, with a standard deviation of .18. Compute and interpret the 95% confidence interval of the mean mpg of the Guzzlers manufactured at that plant. 1) 2) What is a? .05 What is ta? 2.26 (df= 9) Confidence intervals 100 * (1 a )% Confidence interval of x x (ta )( sx ) Another example: A random sample of 10 “Guzzler” SUV‟s produced at Factory K has a mean mpg of 5, with a standard deviation of .18. Compute and interpret the 95% confidence interval of the mean mpg of the Guzzlers manufactured at that plant. 1) 2) 3) What is a? .05 What is ta ? 2.26 (df=9) What numbers do we plug into the formula? Confidence intervals 100 * (1 a )% Confidence interval of x x (ta )( sx ) Another example: A random sample of 10 “Guzzler” SUV‟s produced at Factory K has a mean mpg of 5, with a standard deviation of .18. Compute and interpret the 95% confidence interval of the mean mpg of the Guzzlers manufactured at that plant. 1) 2) 3) What is a? .05 What is ta ? 2.26 (df=9) What numbers do we plug into the formula? 95 % CI of x 5 2.26 * (.18 / 9 ) 4.86 to 5.14 Confidence intervals 100 * (1 a )% Confidence interval of x x (ta )( sx ) Another example: A random sample of 10 “Guzzler” SUV‟s produced at Factory K has a mean mpg of 5, with a standard deviation of .18. Compute and interpret the 95% confidence interval of the mean mpg of the Guzzlers manufactured at that plant. 1) 2) 3) What is a? .05 What is ta ? 2.26 (What are the degrees of freedom?) Plug the numbers into the formula and solve: 95 % CI of x 5 2.26 * (.18 / 9 ) 4.86 to 5.14 Interpret: we are 95% confident that the mean mpg of the Guzzlers produced at Factory K is between 4.85 and 5.15. Confidence intervals around proportions The same procedures can be used to compute a confidence interval for a proportion, so long as the N is greater than 5/p(smaller). Just remember that PQ P(1 P) sP N N Confidence intervals around proportions Example: based on a random sample of 600 voters, 56% say they plan to vote for Obama in the next election. Compute and interpret the 95% confidence interval for the proportion of voters who will vote for Obama. 1) 2) a .05 ta = 1.96 3) 95% CI of x .56 1.96 * ( .56 * .44 ) .52 to .60 600 Interpretation: we can be 95% sure that between 52% and 60% of the vote will go to Obama. (In other words, the poll has a 95% “margin of error” of ±4 percentage points.) Confidence intervals around proportions Another example: in a random sample of 200 Super Bowl viewers, 88% say they agree with the statement: “The halftime show should be abolished altogether, I want to see football, not lame pop stars.” Compute and interpret a 95% confidence interval for the percentage of Super Bowl viewers who agree with the statement. First, check to make sure that the sample size is large enough relative to p(smaller): Is (.12)*200 greater than or equal to 5? Yes! (it equals 24) Therefore, we can use the t distribution to derive a confidence interval. 1) 2) a .05 ta = 1.96 3) 95% CI of x .88 1.96 * ( .88 * .12 ) .835 to .925 600 Interpretation: we can be 95% sure that between 83.5% and 9.25% of Super Bowl viewers want the halftime show to be abolished. (In other words, the poll has a 95% margin of error of ±4.5 percentage points.) Confidence Intervals: Wrapping Up Remember: a confidence interval tells you the precise interval in which the population mean falls with a precise degree of probability. We obtain confidence intervals when we are interested in the true population mean on a particular variable and we only have sample data available. Adopting a given level of confidence, we are always taking a risk of reaching an incorrect conclusion. Make sure you know how to compute a confidence interval, given the sample mean, sample standard deviation, sample size, and a. Make sure you know that to apply the formula to a population proportion (rather than a mean on an interval variable) you only need to know P, the sample size, and a. For example, for a .05 we will identify “correct” confidence intervals 95 times out of 100 (on average) and incorrect ones 5 times out of 100. And that in order to this, it must be true that P(smaller)*N≥5. Let‟s do a few sample problems… Hypothesis testing Another type of inference involves testing explicit hypotheses about some parameter in a population using sample data. We also use the logic of partitioning areas under a sampling distribution to test hypotheses in this fashion. To see how this is done, we‟ll learn how to test hypotheses about the mean in a population. Then we‟ll extract some general steps in hypothesis testing that will apply when we test other types of hypotheses. Hypothesis tests about one mean We begin with a research hypothesis that involves a claim about a mean. We should be able to express that hypothesis in terms of the mean‟s relationship to some target value. Some examples: We know that the mean on variable x in group A equals z and we hypothesize, based on our theory, that the mean in group B differs from the mean in group A. Hypothesis: xB @ z. We know that mean on x in the population from some past time period P equals k and we hypothesize, based on our theory, that the mean has increased (or decreased) since then. Hypothesis: xB > k. Hypothesis: xB < k. Hypothesis tests about one mean Some more examples: We want to test whether the mean in the population exceeds some “ideal” target value, i. (For example, in quality control procedures). Hypothesis: x > i. We know that the population mean on x is r and we want to test whether our sample is representative with respect to x. Hypothesis: x @ r. Hypothesis tests about one mean We can tests hypotheses about whether a population mean equals a specific value using the t-distribution. Essentially, we ask the question: if the population mean equals the target value, x0 , then what is the probability of obtaining a sample mean that differs from x0 by x x0 or more? IF that probability is very small, we can reject the hypothesis that x = x0. IF that probability is substantial, we cannot reject the hypothesis that x = x0. Hypothesis tests about one mean Remember: IF that probability is very small, we can reject the hypothesis that x = x0. IF that probability is substantial, we cannot reject the hypothesis that x = x0. How small is “very small”? How large is “substantial”? Here we rely on convention and tradition. For example, in the social sciences it is customary to treat the threshold probability as .05. That is, we adopt a significance level of .05 by convention. IF that probability < .05, we reject the hypothesis that x = x0. IF that probability > .05 do not reject the hypothesis that x = x0. Hypothesis tests about one mean We call the hypothesis that the population mean equals a particular value the statistical hypothesis (more commonly known as the „null‟ hypothesis). We always test the statistical hypothesis using our data. Remember: a statistical hypothesis always involves an equality. If we reject the statistical hypothesis, we then rule in favor of an alternative hypothesis, which is usually the hypothesis we get from our theory. If we do not reject the statistical hypothesis, we must rule against the alternative hypothesis (but that does not mean we actually “accept” the statistical hypothesis – notice the subtle difference! We test the statistical hypothesis using what we know about the t-distribution. Hypothesis tests about one mean To see how it works, we have to work with a t distribution on the board. We can think of the process of testing a statistical hypothesis about one mean in several different ways. 1) What is the probability of obtaining the sample mean we did if the statistical hypothesis is true? Is it less than .05? If yes, reject the statistical hypothesis. If no, then fail to reject the statistical hypothesis. Hypothesis tests about one mean To see how it works, we have to work with a t distribution on the board. We can think of the process of testing a statistical hypothesis about one mean in several different ways. 2) How far from the target mean does our sample mean have to be in the original scale of x in order to reject the statistical hypothesis at a = .05? Is the sample mean that far from the target mean? If yes, reject the statistical hypothesis. If no, then fail to reject the statistical hypothesis. Hypothesis tests about one mean To see how it works, we have to work with a t distribution on the board. We can think of the process of testing a statistical hypothesis about one mean in several different ways. 3) How far from the target mean does our sample mean have to be in units of t in order to reject the statistical hypothesis at a = .05? (I.e., what is the “critical value” of t, ta, at a = .05?) Does the absolute value of the sample t statistic (the sample mean transformed into units of t) exceed |ta|? If yes, reject the statistical hypothesis. If no, then fail to reject the statistical hypothesis. Example: two-tailed test about one mean Problem: You are a faculty member in the Statistics Department at Quantoid State University and your department chair has assigned you the task of assessing whether the qualifications of this year's crop of grad students differ from the qualifications of last year‟s. Department records show that last year the entering grad students had a combined score on Statistical Competence Test (SCT) of 12.0. You have limited time, so you randomly choose the files of 20 of this year‟s students and determine that their mean SCT score is 13.4 and the variance is 10.0. Example: two-tailed test about one mean Step 1: State the statistical hypothesis and the alternative hypothesis. Statistical Hypothesis: 12.0. Alternative Hypothesis: @ 12.0. Step 2: Describe the sampling distribution. This is a test about a single mean in a population – the population of this year‟s graduate students. Therefore, the sampling distribution is a tdistribution. Example: two-tailed test about one mean Step 3: State the level of significance (a) and determine the critical value of the test statistic. We set a .05 because that is the convention in social sciences. We determine that |t.05| for 19 degrees of freedom is 2.093. Step 4: Observe the sample values and compute the test statistic. t x 0 sx 13.4 12.0 198 . n1 10 19 Example: two-tailed test about one mean Step 5: Compare the test statistic (the sample-based statistic) to the critical value and make the decision about the statistical hypothesis. 1.98 < 2.093. Therefore, we fail to reject the statistical hypothesis. We must reject the alternative hypothesis. Step 6: Interpret the results in plain English. The data provide no evidence that the qualifications of this year‟s graduate students differ from the qualifications of last year‟s graduate students. Example: two-tailed test about one mean What if the mean in the sample of 20 students was 10.5 rather than 13.4? t x 0 sx 10.5 12.0 2.12 2.12 n1 10 19 Example: two-tailed test about one mean What if the mean in the sample of 20 students was 13.8 rather than 13.4? t x 0 sx 138 . 12.0 2.55 n1 10 19 Example: one-tailed test about one mean Problem: You are a faculty member in the Statistics Department at Quantoid State University and all your colleagues say that this year‟s crop of grad students is clearly better qualified than last years. You decide to test this rumor statistically. Department records show that last year the entering grad students had a combined score on Statistical Competence Test (SCT) of 12.0. You have limited time, so you randomly choose the files of 20 of this year‟s students and determine that their mean SCT score is 13.4 and the variance is 10.0. Example: one-tailed test about one mean This is what we call a one-tailed research question. The research question specifies either that the population mean is greater than some value (in this case, 12.0) or that the population mean is less than some value. In this case, we are willing to rule out a priori that the population mean falls in the opposite tail. In our example, we are willing to rule out the possibility that this year‟s cohort of grad students is less qualified than last year‟s When using a one-tailed test, you must keep track of the signs of the critical value of t and the test statistic, and you must make sure to look up the one-tailed ta rather than the two-tailed ta in Statistical Table C. Example: one-tailed test about one mean Step 1: State the statistical hypothesis and the alternative hypothesis. Statistical Hypothesis: 12.0. Alternative Hypothesis: > 12.0. Step 2: Describe the sampling distribution. This is a test about a single mean in a population – the population of this year‟s graduate students. Therefore, the sampling distribution is a tdistribution. Example: one-tailed test about one mean Step 3: State the level of significance (a) and determine the critical value of the test statistic. We set a .05 because that is the convention in social sciences. We determine that t.05 for 19 degrees of freedom using a one-tailed for a positive mean is +1.729. Step 4: Observe the sample values and compute the test statistic. t x 0 sx 13.4 12.0 198 . n1 10 19 Example: one-tailed test about one mean Step 5: Compare the test statistic (the samplebased statistic) to the critical value and make the decision about the statistical hypothesis. 1.98 > 1.729. Therefore, we reject the statistical hypothesis. We accept the alternative hypothesis. Step 6: Interpret the results in plain English. The data provide evidence that the qualifications of this year‟s graduate students are better than the qualifications of last year‟s graduate students. One-tailed vs. two-tailed tests Use a two-tailed test when? When there is no basis to specify in advance whether μX > μ0 or μX < μ0 . When the research question specifies an inequality or a difference between the population mean and some value without suggesting that the difference is positive or negative One-tailed vs. two-tailed tests Use a one-tailed test when? When we can rule out in advance either that μx > μ0 or μx < μ. For example, say we know the mean population height for men is 5'10". We measure the height of 100 randomlyselected women, thereby obtaining a sample mean height of 5'6" and standard deviation of 0.4". Should we use a onetailed or two-tailed test to determine whether the average height of women equals that of men (5'10")? One-tailed vs. two-tailed tests Which tail do you look at in a one-tailed test? Ask: Do we think that μx > μ0 or that μx < μ0 ? If we think μx > μ0 then the "critical region" or "red zone" (where we reject the null hypothesis that μx = μ0 ) must be in the right-hand tail, so take the positive value of tα . If we think μy < μ0 then the "critical region" or "red zone" (where we reject the null hypothesis that μx = μ0 ) must be in the left-hand tail, so take the negative value of tα . One-tailed vs. two-tailed tests Remember: You must specify in advance whether you are using a one-tailed or two-tailed test. And, if you are using a one-tailed test, you must specify the appropriate tail in your alternative hypothesis. The main reason for this is that it is easier to reject a statistical hypothesis using a one-tailed test (as our example showed). If you don‟t specify a one-tailed test in advance, you might be accused of “cheating”!! Some basic principles in hypothesis testing about one mean The greater the significance level (the higher a), the smaller ta and the easier it is to reject the statistical hypothesis. The greater the difference between the target mean and the sample mean, the larger the test value of t and the easier it is to reject the statistical hypothesis. The greater the sample size and the smaller the sample variance, the larger the test value of t and the easier it is to reject the statistical hypothesis. General steps in hypothesis testing Step 1: State the statistical hypothesis and the alternative hypothesis. Step 2: Describe (and draw)the sampling distribution. Step 3: State the level of significance (a) and determine the critical value of the test statistic. Step 4: Observe the sample values and compute the test statistic. Step 5: Compare the test statistic (the sample-based statistic) to the critical value and make the decision about the statistical hypothesis. Step 6: Interpret the results in plain English. General steps in hypothesis testing These same steps are used in a wide range of statistical tests. So far, we have learned only one statistical test: a t-test about the mean in one population. We will learn other tests, including: A t-test comparing the means in two populations, A chi-square test evaluating whether two nominal variables are statistically independent (or statistically related), A t-test assessing whether a regression coefficient equals zero in a population. For now, let‟s go over some sample problems involving tests about one mean…