Business Statistics. Lecture 10: Course Review
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1 Business Statistics Lecture 10: Course Review 1
2 Descriptive Statistics for Continuous Data Numerical Summaries Location: mean, median Spread or variability: variance, standard deviation, range, percentiles, quartiles, interquartile range Graphical Descriptions Histogram Boxplot Scatterplot 2
3 Descriptive Statistics for Categorical Data Numerical Measures: Mode: most commonly occurring value Frequency table: how often each value occurs Graphics: Bar chart of frequencies (histogram) Mosaic chart (stacked bar chart) Pareto chart 3
4 Basic Probability Rules: Probability of the union of two disjoint events: Pr(A or B) = Pr(A U B) = Pr(A) + Pr(B) In general, probability of the union of two events: Pr (A or B) = Pr(A U B) = Pr(A) + Pr(B) Pr(A B) Complimentary events: Pr(not A) = Pr(A c ) = 1 - Pr(A) Independent events: Pr(A and B) = Pr(A B) = Pr(A) x Pr(B) Independent versus dependent events 4
5 The Normal Distribution Symmetric Bell shaped Unimodal Thin tails 5
6 Why the Normal Distribution? Normal distribution describes many natural phenomenon well Central Limit Theorem: Distribution of sums of random variables tends toward the normal The more things that are summed, the more like the normal Result is that averages tend to have a normal distribution 6
7 The Empirical Rule If the normal distribution fits well then: 68% of the data is within 1 SD of the mean 95% within 2 SD 99% within 3 SD % % 99% Z 7
8 Standardizing Standardizing means turning an observation from a N(, 2 ) into a N(0,1) observation If X comes from a N(, 2 ) then X Z has a N(0,1) distribution If and are estimated, then use X x Z s 8
9 Remember: Statistics versus Parameters A statistic is a numerical summary of data Statistics can be for samples or populations and s are examples of sample statistics X and are parameters of the normal distribution We often estimate parameters with statistics Estimate Estimate with X with s 9
10 The t Distribution 0.40 normal 0.30 T3 T10 T Z= number of SE s from the mean 10
11 Degrees of Freedom (df) The more degrees of freedom we have, the better we can estimate The better we estimate, the closer we are to being known Thus, the more df we have, the closer t values are to z values Calculating degrees of freedom: Each observation adds one degree of freedom One degree of freedom is used up when we calculate X There are n-1 degrees of freedom left 11
12 What is a Sampling Distribution? A sampling distribution is a probability distribution of a sample statistic 2 For example, if X ~ N(, ) then X ~ N, n 2 The standard deviation of is called the standard error of the mean For a sample of size n, standard error of the mean is 1/ n times the standard deviation of an individual observation X 12
13 Picturing a Sampling Distribution Individual Mean of Distribution of individual observations: standard deviation= ShaftDiam Sampling distribution of the mean Distribution of the sample mean for samples of size n=5: X 5 13
14 Generating a Sampling Distribution Process that generates random Xs Take and plot Xs individually Take 5 Xs, average them, and plot Moments Mean Std Dev Std Err Mean upper 95% Mean lower 95% Mean N Means almost exactly the same Moments Mean Std Dev Std Err Mean upper 95% Mean lower 95% Mean N Averages much less variable; there is a specific relationship! 14
15 Individual Mean of 5 Unobserved pop mean The Main Idea of Confidence Intervals Because of the CLT, we know that X is within 2 SE s of 95% of the time Alternatively, is within 2 SE s of 95% of the time Sample mean X (Unobserved) dist. of sample mean 95% confidence interval for pop mean ShaftDiam (Unobserved) dist. 15 of population
16 Calculating Confidence Intervals for the Mean The formula: Example Sample mean: = Sample standard deviation: s = Sample size: n = 100 So, t Then: x t s n x n1, / 2 / x s / n 99, / 100 [ , ] 16
17 How Confidence Intervals Behave Width of CI s: w 2t n1, / 2 s Margin of error: E tn1, / 2 n s n Bigger SD Bigger SE wider intervals Bigger sample size Smaller SE narrower intervals Smaller t values narrower intervals Higher confidence Bigger t values wider intervals 17
18 t vs. z Use t when you don t know The t distribution assumes the data are normally distributed Options if data are not normally distributed: Transform the data (logarithms) If transformations don t work and sample size is big ( > 30) ignore the problem If transformations don t work and sample size is small, read the book about nonparametric tests 18
19 Surveys Surveys and Sampling Random selection ensures survey is representative Randomized surveys can be generalized to population Types of sampling Bias vs. variance Power calculations Confidence intervals for proportions 19
20 Hypothesis Testing Start with a theory or hypothesis For example, = 0 Collect some data Ask: How unusual is it to see this data if the null hypothesis is true? If it s unusual, reject the null hypothesis If not, fail to reject the null Remember, determine the hypothesis to be tested before looking before looking at the data 20
21 Ties Back to the Empirical Rule 68% 95% Z If we hypothesize that the data come from a N(0,1) distribution, how unusual an observation must we see to reject our hypothesis? It depends on the alternative hypothesis 21
22 For Example, a Two-sided Test Null: The mean is equal to zero (H 0 : = 0) Alternative: The mean is not equal to zero (H a : 0) If the rejection criterion is p-value < 0.05, we reject if our observation is greater than 1.96 or less than -1.96: 68% 95% Z 22
23 Interpreting p-values Small p-values mean either the null is false or that a rare event happened If the process mean is actually 0, then we would see a sample mean greater than 2 or less than -2 about 1 time in 20 So, if we see a mean between -2 and 2, we conclude that the process mean is not different from 0 Otherwise, we conclude the alternative that mean is not equal to 0 23
24 z-tests vs. t-tests As with confidence intervals, if is known, then do z-test Based on the normal distribution If we must estimate by s, then to a t-test Uses the t distribution Since is almost never known in the real world, JMP defaults to the t-test The only difference is which distribution is used to calculate the p-value 24
25 Null Hypothesis: x - y =0 Test Statistic: Estimated Standard Error: Rescaled Test Statistic: 25 Y X Comparing Two Means 2 2 y x x y s s n n 2 2 y x x y X Y t s s n n
26 One-sample and Two-sample Tests In a one-sample test of, choose * Then T = X, so the test statistic is * * * T X X t s. d.( T) s. e.( X ) s n In a two-sample test, you re often testing whether the means are equal T = X Y, and the test statistic is * 2 T ( X Y ) 0 sx t ( X Y ) s. d.( T) s. e.( X Y ) n x s n 2 y y 26
27 Two-sample vs. Paired Tests Two sample t-test requires independence between two samples Paired t-test assumes two observations taken for each unit in the sample Allows observations to be dependent Observations on the same unit likely to be more similar than obs ns on different units Good news: under these conditions, paired t-test more powerful 27
28 Paired t-test Looks at Differences x 1 -y 1 =d 1 x 2 -y 2 =d 2. x n -y n =d n Calculate differences for each observation Calculate sample mean and SD of differences Do a one-sample t-test for differences: H 0 : mean difference is zero H a : mean difference is not 0 28
29 Terminology One-sided vs. two-sided Comes from the statement of the alternative hypothesis Are you calculating the p-value using one tail or two? One-sample vs. two-sample Comes from the type of data and the question you are answering Are you testing a mean or a difference between means? 29
30 Which Test? How many populations are sampled? One: one-sample test Two: read on Are observations in first sample independent of observations in second sample? Yes: two-sample t-test No: paired t-test Big Clue: Paired t-test needs two observations from each unit Unequal sample sizes 2 sample test Equal sample sizes you have to decide 30
31 Correlation A measure of the strength of the linear relationship between X and Y Xs and Ys are two different (continuous) variables observed on the same units in your sample Correlation (r) close to: +1: strong positive linear relationship 0: no linear relationship -1: strong negative linear relationship 31
32 Pizza Sales ($000) Estimating the Linear Relationship Correlation measures the strength of the linear relationship between X and Y Estimating the actual linear relationship is given by the regression of Y on X yˆ aˆb ˆ x intercept slope Income ($000) yˆ x 32
33 Linear Model General expression for a linear model y a bx e i a and b are model parameters e is the error or noise term Error terms often assumed independent 2 observations from a N(0, ) distribution i 33
34 Estimating the Linear Model Given some data we will estimate the regression model parameters (a and b) with coefficients: where ŷ yˆ aˆb ˆ x i y, x y y is the predicted value of y, n, x x n i 34
35 Minimizing Sum of Squares yˆ aˆbx ˆ i error y yˆ i i i i For each observation in the data set, your line predicts where Y should be The residual from i th data point is how far the true Y value is from where the line predicts SE e e e line 1 2 n The sum of squared residuals (or sum of squared errors) gives an overall measure of how well the line fits Choose a and b to make SE line as small as possible 35
36 Coefficient of Variation (R 2 ) Some of the variation in Y can be explained by variation in X and some cannot R-squared tells you the fraction of variance of Y that can be explained by X R SE 1 1 SE 2 line av y y i i yˆ y i i
37 JMP Regression Output 37
38 What We Have Learned in this Course Descriptive statistics A bit about probability Inference for a population mean Confidence intervals Hypothesis testing One-sample tests Two-sample tests Paired tests Introduction to simple linear regression 38
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