University:
Eastern Kentucky University
Course:
STA 270 | Applied Statistics
Academic year:

2022
Views:

48

Pages:

31
Author:

sommatosmby

Relational Symbols = equals, is the same as ≠ is not equal to, is different from > is greater than, is more than, exceeds, is above ≥ or >= is greater than or equal to, is at least, is not less than < is less than, is fewer than, is below ≤ or <= is less than or equal to, is at most, does not exceed, is not greater than, is no more than AZ score -> calc-> normal-> select between-> enter confidence level in decimal-> compute-> positive z value Estimating a Population Mean The assumptions for this confidence interval procedure are: α is unknown AND x is normally distributed Since is σ unknown, if we replace σ with s, we are basing our estimate of μ on yet another estimate. This adds extra error to the process and we need a distribution that will account for this extra error. StatCrunch Stat, T stats, One sample If the confidence level and margin of error are specified in advance, we can calculate the sample size necessary to meet those criteria . Note that this formula uses Z scores, not T. Generally, since we don’t know n, (it’s what we’re trying to find!) we don’t know our df and therefore which T distribution to use. HOWEVER, StatCrunch has a procedure to find a sample size for estimating a mean using either T or Z. StatCrunch Stat, Z stats, (OR T stats) One sample, Power/sample size The Language of Hypothesis Testing ➢ Hypothesis testing always starts with two hypotheses. The Null Hypothesis (Ho) – reflects the established value of the parameter. It is assumed to be accurate until we have evidence otherwise. The Alternative Hypothesis (Ha) – is the challenger. It questions whether the established parameter is still accurate or not. For a hypothesis test of one population parameter, the null hypothesis, Ho, has the form Ho: parameter = number (Example Ho: µ = 12.3) There are two types of potential error in a hypothesis test. 1. Type I error - rejecting Ho when Ho is true 2. Type II error - not rejecting Ho when Ho is false A type 1 error is also known as a false positive and occurs when a researcher incorrectly rejects a true null hypothesis. This means that your report that your findings are significant when in fact they have occurred by chance. The probability of making a type I error is represented by your alpha level (α), which is the p-value below which you reject the null hypothesis. A p-value of 0.05 indicates that you are willing to accept a 5% chance that you are wrong when you reject the null hypothesis. Mean = n times p Standard deviation = n times p, times 1 minus p, then do the square root The Significance Level - is the probability of making a type I error. It is denoted by α Hypothesis Tests for a Population Proportion We will learn three different methods for testing hypotheses. 1. The critical value, or traditional approach 2. The p-value approach 3. The confidence interval approach Test statistic – The statistic we use to measure the distance between the sample statistic and the established parameter in terms of standard deviation. ➢ Every hypothesis test has a test statistic, each of which is basically the same for each test. (statistic – parameter)/(standard error) Rejection Region – The set of values for the test statistic that leads to rejection of the null hypothesis. Non-rejection Region – The set of values for the test statistic that leads to non-rejection of the null hypothesis. Critical Value/value(s) - Z or T scores that separate the rejection and non-rejection regions. p-value (probability value) - The probability of observing a test statistic more extreme than the one observed if the null hypothesis is true. Summary: If the p-value ≤ α , reject Ho If the p-value > α , do not reject Ho You can think of it this way, “p-value low, the Ho must go” StatCrunch Stats, Proportion stats, One sample Hypothesis Tests for a Population Mean, μ For a hypothesis test of one population mean, the hypotheses are: The assumptions for this test are: 1. x is normally distributed 2. The population standard deviation, sigma ( σ ) is unknown StatCrunch Stat, T stats, One sample Inferences on Two Samples Inference for Two Population Proportions KNOW: For inferences of two population proportions, we are estimating the DIFFERENCE of the population proportions, p1 − p2 . ➢ When we test ONE parameter, we test it against a number. When we test TWO parameters, we test them against each other. StatCrunch Stat, Proportion stats, Two sample Inference About Two Means - Dependent Samples & Independent Samples Dependent Independent ✦ μd The mean of the differences ✦ μ1- μ2 The difference of the means d = difference between paired data. ➢ Inferences for two samples that are dependent are called “Matched Pairs”. In a hypothesis test for matched pairs, we are testing the mean of the differences which is denoted . The differences are individually denoted by d. The data values we test are these differences. The assumptions are: 1. The data are random 2. The data are matched pairs 3. The differences are normally distributed 4. The differences are independent StatCrunch Stat, T stats, Paired Here, we will estimate the difference between two independent population means, m1 − m2. The assumptions are: 1. Independent and random samples (no outcome/sample affects the otheroutcomes/sample) 2. Each x is normally distributed Which is a t test with the smaller of n1-1 or n2-1 degrees of freedom. StatCrunch Stat, T stats, Two sample Inference on Categorical Data Goodness of Fit Test In general, the hypotheses are: Ho: The data follows (some) distribution H1: The data does not follow (that) distribution Keep in mind that: Oi represents the observed counts in category i. Ei represents the expected counts in category i. k represents the number of categories. n represents the number of independent trials of an experiment. Ei = npi where pi is the probability of observing outcome i. The assumptions are: 1. All the expected frequencies are greater than or equal to 1. 2. No more than 20% of the expected frequencies are less than 5. If these assumptions are met, then this test follows a Chi-square distribution and is always right-tailed. StatCrunch Stat, goodness of fit, Chi-square test. The “observed” will be the data, the “expected” will be the total number of observed multiplied by the corresponding probability for each category. Comparing Three or More Means( ANOVA) The variability among the sample means is called between-sample variability, and the variability of each sample is the within-sample variability. One-way Analysis of Variance We will extend the concept of comparing 2 population means to comparing 3 or more population means. This procedure is known as “analysis of variance” or ANOVA. The assumptions are: 1. Simple random samples come from each of the k populations (or the subjects are randomly assigned to a group) 2. All k samples are independent 3. Each population is normally distributed 4. The populations have equal variances* *Assumption 4 can be difficult to determine. A rule of thumb is that the largest standard deviation can’t be more than twice the smallest standard deviation. If the assumptions are satisfied, this is an F test and follows the F distribution. The analysis of “variance” part comes in the F test statistic, which is a ratio of two variances. • The numerator is a measure of variance in sample means from one treatment group to the next, which is denoted “MST.” denominator is a measure of sample variance from each treatment group which is then weighted and • The averaged, which is denoted “MSE.” Both MSE and MST are “unbiased” estimators of 2 , the population variance. What this means is that if H0 is “true”, the ratio of MST/MSE is pretty close to 1. If H0 is not true, at least one of the sample means must be “far away” from the mean of the entire data set which causes MST to be large relative to MSE, which results in an F statistic significantly larger than 1, which leads to the rejection of H0. So that you know the relationships of the calculations to the test statistic, here is an example of an ANOVA table. Keep in mind k is the number of groups/treatments, n is the total number of data values. StatCrunch Stats, ANOVA, one way ➢ The Describing the Relationship between Two Variables best predictor of longevity is… do you have a guess? EDUCATION! Scatter Diagrams and Correlation Bivariate data – Two measured outcomes from a single subject/item/source which are written as ordered pairs, (x, y) where x is the predictor variable and y is the response variable. Scatter diagram/plot – A graph of bivariate data for which the value of the predictor variable, x is represented on the horizontal axis and the value of the response variable, y is represented on the vertical axis. The linear correlation coefficient, denoted r – measures the strength of the linear relationship between the paired sample data. −1 r 1 always and r has no units. (miles, gallons, speed…none of that) • The closer r is to -1, the stronger the negative linear relationship/association. r = -1 means that there is a perfect, negative, linear relationship between x and y. (Which only exists in fairy tale land) • The closer r is to 1, the stronger the positive linear relationship/association. r = 1 means that there is a perfect, positive, linear relationship between x and y. (And it only exists in fairy tale land, too) (positive or negative) the worse the linear relationship/association. Scatterplots with StatCrunch Graph, Scatter plot The correlation coefficient with StatCrunch Stat, regression, simple linear Least-Squares Regression • The closer r is to 0, Residuals - The vertical distances between each data value and the regression line. They represent how close the prediction (regression line) comes to the actual observed y-value. Because of this, residuals are often called “errors”. (It’s statistics! There’s always error… and that’s okay) The “best” line, called the least squares regression line is the one that most accurately describes a set of data and minimizes the sum of the squared residuals, or ∑ 2 . This is called the “least squares criterion.” ANY other line will have a sum of squared residuals (or total error) that is larger. A single residual can be found by: residual = observed y (from the data/ordered pair) minus predicted y (from evaluating the regression line at the given value of x). We can also find individual residuals using StatCrunch. Extrapolation – When one tries to use a regression line to make predictions outside of the range of the observed values. Never a good idea. The slope and y-intercept of the regression line and the residuals with StatCrunch Stat, regression, simple linear In the “simple linear regression” set-up window, click “residuals” under the “Save” menu about halfway down that window. Diagnostics on the Least-Squares Regression Line The differences between any two of these three things are called “deviations.” • the observed data values • the regression line • the mean of y or y-bar The Coefficient of Determination, r  2 – The percentage  oftotal  error eliminated by using (the regression line) to predict a value of y instead of (the average y). It can also be thought of as how well the regression line describes the relationship between the x- and y-variables. 0 ≤ 2 ≤ 1 always! Interpreting the coefficient of determination: Step 1: convert r-square value from a decimal to a % by X 100. Step 2: use this format for R-square (coefficient of determination) problems: ____% of the variation/error in estimating _(y-axis) can be explained by the least squares regression line/_(x-axis). “What Proportion of the variability”= they want the R2 With a residual plot, you want no discernible pattern, showing constant error variance. Influential Observations – significantly alter the regression line’s slope and/or y-intercept when they are included/excluded.  Residual plots – Give additional information of whether a good, reliable, linear relationship exists between the two variables. If a residual plot shows a pattern, it means that the data is not random/independent. Must be independent and random!!! Can determine this by constructing a residual plot in Statcrunch The coefficient of determination and residual plots (x, residual) with StatCrunch Stat, regression, simple linear “Residuals vs. x-values” in the “Simple linear regression” setup menu under “Graphs” for a residual plot Inference on the Least-Squares Regression Model ➢ In sections 14.1 and 14.2, we’ll use hypothesis testing and confidence intervals to make inferences for a regression model. Testing the Significance of the Least-Squares Regression Model The least-squares regression equation is: The values of b1 and b0 are calculated from sample data so they’re statistics. If we had population data, the regression equation would be Recall that each sample statistic has a distribution. (shape, mean, standard deviation) Since b1 and b0 are statistics, they each have a sampling distribution with which we would use to perform inference. We’ll focus on the distribution of b1 , the slope. The sampling distribution for b1 occurs when one of the predictor variables, x, is held fixed and the observed outputs are repeatedly sampled. Each of these distributions is normal with equal standard deviations, but since this standard deviation must be estimated from the sample data, test statistics and confidence intervals will follow a t distribution with n-2 df. (df = 2 since two parameters are being estimated) The standard error of the estimate - is the standard deviation of y for any given value of x. It’s estimated by Se . In the StatCrunch regression output, it’s called “Estimate of error std. Dev.” One assumption for inference on the least-squares regression model is: For any specific, fixed value of the explanatory variable x, the mean of the corresponding responses in the population depends linearly on x, or ➢ We check this assumption with a plot of the residuals against x, the explanatory variable. We learned how to do this in 4.3. If the plot shows a pattern, then a linear model is not appropriate. ➢ We check this assumption by verifying that the residuals are normally distributed. To do so, you can save the residuals from the regression set-up menu and then run a normal probability plot on them, or use the directions above, in bold. Because most (or all) of the y’s from our sample data don’t lie directly on the regression line, we learned that the distances between the two are called residuals. Here, the residuals are denoted by � with the i th corresponding to the i ordered pair in the sample. When we add that error term to the regression model/equation, the complete least-squares regression model is given by Do the sample data provide sufficient evidence to conclude that a linear relation exists between the two variables? In short, is the slope meaningful? The hypotheses generally are : ➢ We want to reject H0! If we can’t reject Ho, then we use y , the mean response to estimate y for any value of x. The results of this test are in the regression output, by default. StatCrunch In the regression menu, “Graphs”, click “QQ plot of residuals” StatCrunch Stat, regression, simple linear, click in front of “confidence intervals” “Reject H0 and conclude that a linear relation exists between the _______and _______ at the α=0.05 level of significance.” Confidence and Prediction Intervals 9 9 The 9 9 9 regression 9 9 9 9 9 9 9 9 equation 9 from the diamond data is = 11598. − 2298.4where y ˆ is 9 the predicted price for a given weight, x. This actually has TWO interpretations! 1) It represents the mean price for all diamonds whose weight is x. 2) It represents the predicted price for a randomly selected diamond whose weight is x. Ifwe let x=0.7,  we  get = 11598.9(0.7) − 2298.4= 5820.83 The two interpretations would be: 1) The mean price for all diamonds with a weight of 0.7 is 5820.83 2) The predicted price for a randomly selected diamond with weight 0.7 is 5820.83 Of course, there’s always error in making such predictions, so we can construct intervals about any predicted value to describe its accuracy. The type of interval will depend on what we’re predicting, 1 or 2, above. 1) A CI for ,  the  mean response of all  y for a specific x is a confidence interval for a mean response. 2)A  CIfo  predicted response for a specific x is a prediction interval for an individual response. The assumption for either of these intervals is that the residuals are normally distributed or the sample size is large. StatCrunch Stat, Regression, simple linear. Click in front of “Confidence intervals” for a confidence interval for a mean response. Enter a “Prediction of y” for a prediction interval. Confidence intervals for a mean response: Intervatals construction about the predicted value of y at a given level x. These intervals are used to measure the accuracy of the MEAN RESPONSE OF ALL INDIVIDUALS IN THE POPULATION

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