Transcript Lecture 5
Review • Measures of Central Tendency – Mean, median, mode • Measures of Variation – Variance, standard deviation Variance is defined as 2 s = Σ ( X - X̄ ) N 2 The Normal Curve • The mean and standard deviation, in conjunction with the normal curve allow for more sophisticated description of the data and (as we see later) statistical analysis • For example, a school is not that interested in the raw GRE score, it is interested in how you score relative to others. • Even if the school knows the average (mean) GRE score, your raw score still doesn’t tell them much, since in a perfectly normal distribution, 50% of people will score higher than the mean. • This is where the standard deviation is so helpful. It helps interpret raw scores and understand the likelihood of a score. • So if I told you if I scored 710 on the quantitative section and the mean score is 591. Is that good? • It’s above average, but who cares. • What if I tell you the standard deviation is 148? • What does that mean? • What if I said the standard deviation is 5? • Calculating z-scores Converting raw scores to z scores What is a z score? What does it represent Z = (x-µ) / σ Converting z scores into raw scores X=zσ+µ Z = (710-563)/140 = 147/140 = 1.05 Finding Probabilities under the Normal Curve So what % of GRE takers scored above and below 710? The importance of Table A Why is this important? Inferential Statistics (to be cont.) Stuff you don’t need to know: pi = ≈3.14159265 e = ≈2.71 The Normal Curve and Sampling A. A sample will (almost) always be different from the true population B. This is called “sampling error” C. The difference between a sample and the true population, regardless of how well the survey was designed or implemented D. Different from measurement error or sample bias Sampling distribution of Means • The existence of sampling error means that if you take a 1000 random samples from a population and calculate a 1000 means and plot the distribution of those means you will get a consistent distribution that has the following characteristics: Characteristics of a Sampling Distribution • 1. the distribution approximates a normal curve • 2. the mean of a sampling distribution of means is equal to the true population • 3. the standard deviation of a sampling distribution is smaller than the standard deviation of the population. Less variation in the distribution because we are not dealing with raw scores but rather central tendencies. Why is the normal curve so important? • If we define probability in terms of the likelihood of occurrence, then the normal curve can be regarded as a probability distribution (the probability of occurrence decreases as we move away from the center). • With this notion, we can find the probability of obtaining a raw score in a distribution, given a certain mean and SD. Probability and the Normal Curve In chapter 6 – we are not interested in the distribution of raw scores but rather the distribution of sample means and making probability statements about those sample means. Probability and the Sampling Distribution Why is making probabilistic statements about a central tendency important? • 1. it will allow us to engage in inferential statistics (later in ch. 7) • 2. it allows us to produce confidence intervals Example of number 1: • President of UNLV states that the average salary of a new UNLV graduate is $60,000. We are skeptical and test this by taking a random sample of a 100 UNLV students. We find that the average is only $55,000. Do we declare the President a liar? Not Yet!!!! We need to make a probabilistic statement regarding the likelihood of Harter’s statement. How do we do that? With the aid of the standard error of the mean we can calculate confidence intervals - the range of mean values within with our true population mean is likely to fall. How do we do that? • First, we need the sample mean • Second, we need the standard deviation of the sampling distribution of means (what’s another name for this?) • a.k.a standard error of the mean What’s the Problem? • The problem is… • We don’t have the standard deviation of the sampling distribution of means? • What do we do? First – let’s pretend • Let’s pretend that I know the Standard Deviation of the Sampling Distribution of Means (a.k.a. the standard error of the mean). It’s 3000 • For a 95% confidence interval we multiply the standard error of the mean by 1.96 and add & subtract that product to our sample mean • Why 1.96? • What’s the range? So is President Ashley Lying? CI = Mean + or – 1.96 (SE) = 55,000 +/- 1.96 (3000) = 55,000 +/- 5880 = $49,120 to 60,880 Let’s stop pretending • We Can Estimate the Standard Error of the Mean. – Divide the standard deviation of the sample by √N-1 • Multiply this estimate by t rather than 1.96 and then add this product to our sample mean. • Why t? The t Distribution • Empirical testing and models shows that a standard deviation from a sample underestimates the standard deviation of the true population • This is why we use N-1 not N when calculating the standard deviation and the standard error • So in reality, we are calculating t-scores, not z-scores since we are not using the true sd. • So when we are using a sample and calculating a 95% confidence interval (CI) we need to multiply the standard error by t, not 1.96 • How do we know what t is? • Table in back of book • Df = N - 1 Confidence Intervals for Proportions Calculate the standard error of the proportion: Sp = P1 P N 95% conf. Interval = P +/- (1.96)Sp Example • National sample of 531 Democrats and Democratic-leaning independents, aged 18 and older, conducted Sept. 14-16, 2007 • Clinton 47%; Obama 25%; Edwards 11% • P(1-P) = .47(1-.47) = .47(.53) = .2491 • Divide by N = .2491/531 = .000469 • Take square root = .0217 • 95% CI = .47 +/- 1.96 (.0217) • .47 +/- .04116 or 0.429 to .511 Midterm • Key terms from Schutt chapters 1-5 • Statistical Calculations by hand – Mean, Median, Mode – Variance/Standard Deviation – Z-scores – Standard errors and confidence intervals using z or t