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Statistical Inference Based on a Single Sample Point and Interval Estimation of Parameters Population Parameters: Mean (µ) Variance (2) Subset of the population Sample Statistics: Mean ( ), Variance (s2) is the best point estimate of µ s2 is the best point estimate of 2 In general, a statistic (i.e., a function of the sample) is considered the best estimate of the corresponding population parameter. 100(1- α )% confidence interval: A 100(1- α )% confidence interval for μ is an interval (L, U) constructed in such a way that100(1- α )% of all similarly constructed intervals, in repeated sampling, would contain the true parameter value μ. Understanding the Concept a Confidence Interval Machine 95% Confidence Interval Formula for μ Produces Items (95% good, 5% bad) Randomly select only one item Produces intervals (good: encloses μ 95% of the time, bad: fail to enclose μ 5% of the time) Since 95% of all items are good, we say with 95% confidence that the selected item is a good item. A sample of n values gives only one interval Examples of Some Estimation Problems: Estimate the mean sentence complexity scores of all children. Estimate the mean pulse rate of all U.S. adult males who jog at least 15 miles per week Estimate the mean body temperature of all adults in USA. Estimate the proportion of adults who were victims of a crime. Estimate the proportion of all hourly employees earning minimum wage Since 95% of all intervals constructed by the formula enclose μ, we can say with 95% confidence that the interval obtained from the observed sample 2 encloses μ. Confidence Interval Formula Standard confidence Interval for a parameter θ: (Point Estimate of θ - Margin of error of estimation, Point Estimate of θ + Margin of Error of estimation) Examples (Just for your understanding, will not do any calculations now) What factors/parameters affect the width of CI and how? If you increase n, width will decrease given everything else same; If you increase degree of confidence, width of CI will increase given everything else same; Statistical Inference based on one sample- Tests of Hypothesis ● Instead of estimating parameters, the goal of a test of hypothesis is to see if the data provide sufficient evidence to support a research hypothesis about population parameters (e.g., μ, σ2). ● Elements of a test of hypothesis problem 1. Alternative or Research Hypothesis (Ha) – A statement that contradicts the null hypothesis. It represents researcher's claim about the population parameter. Accept Ha only when data provides sufficient evidence to establish its truth. 2. Null Hypothesis (H0) – A statement about the value(s) of population parameter(s) which we accept as true until proven false. Null (H0) versus Research (Ha) hypotheses – H0 needs no proof; Ha needs proof/evidence/support from data; • A person is innocent is H0 in our court system and needs no proof. • A person is guilty is Ha in a court system and must be proven by evidence; More on Elements of a Test of Hypothesis 3. Test Statistic and its sampling distribution under H0 – A test statistic is a function of sample and H0. For any given sample and null hypothesis, it gives a score that helps to decide whether to reject H0. The decision depends on the sampling probability distribution of the test statistic when H0 is true. 4. Rejection Region (Summary scores for Rejecting H0) – It consists of all values of the test statistic for which H0 is rejected. – This rejection region is selected in such a way that the probability of rejecting true H0 is equal to α (a small number usually 0.05). – The value of α is referred to as the level of significance 5. Conclusion – Reject H0 if the summary score (obtained from test formula) falls in the rejection region (or if the p-value < α). Otherwise, do not reject H0 6. P-value or significance probability – proportion of samples that would be unfavorable to H0 (assuming H0 is true) if the observed sample is considered unfavorable to H0. Remarks: - If the p-value is smaller than α then reject H0. - If the observed test score in step 3 fall in the rejection region, then reject H0 Examples of some test of hypothesis problem ● A consumer group purchases 49 family-size 20-ounce bottles of ketchup , weigh the contents of each, and finds that the mean weight is 19.86 ounces, and s = 0.22 ounces. Do the data provide sufficient evidence for the consumer group to conclude that the true mean fill per family-size bottle is les than 20 ounces? ● A long-range missile missed its target by an average of 0.88 miles. A new steering device is supposed to increase accuracy, and a random sample of 36 missiles were equipped with this new mechanism and tested. These 36 missiles missed by distances with a mean of 0.76 miles and a standard deviation of 0.04 miles. Is there sufficient evidence to conclude that the new steering system lower the mean miss distance? ● A car manufacturer wants to test a new engine to determine whether it meets new air pollution standard (true mean emission must be less than 20 parts per million of carbon). Ten engines manufactured for testing purposes yield the following emission levels with mean =17.57 and std dev = 2.9522: • 15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 19.6, 17.9, 12.7, 14.9 Do the data supply sufficient evidence to allow the manufacturer to conclude that this type of engine meets the pollution standard? Two Types of Errors ● Decision Table and Error Rates H0 is True H0 is False Sufficient evidence to Type I Error reject H0 Correct Decision Insufficient evidence to Correct Decision reject H0 Type II Error Each cell the above 2 by 2 tables is an event; prob(Type I Error) = α ; prob(Type II Error) = β Examples of Cases with Type I Error ● Example 1 (New York Times; January 3, 2011). – Thirty years after Mr. Dupree was imprisoned for rape and robbery, prosecutors in Dallas declared him innocent in light of new DNA evidence. • H0: Mr. Dupree was innocent • Ha: Mr. Dupree was guilty • Conclusion: Court found him guilty – A true null hypothesis was rejected by the court and thus a TYPE I ERROR was committed in this case. Examples of Possible Type II Error ● Burlington Free Press: – Isaac Turnbaugh, of Randolph, Vt., was found not guilty in 2004 of first-degree murder in the killing of Declan Lyons (24) who was shot dead in 2002 while working at the American Flatbread Co., a local pizza restaurant. Turnbaugh, 28, phoned the Randolph police in July and confessed to shooting Lyons in the head with a rifle. – If this Free Press story was right, a type II error was committed by the jury. ● Casey Anthony’s Trial – Casey Anthony was found not guilty. • No Type I error in this case (why?). – If Casey was not the killer, then it was a correct decision. • Type II error? – If Casey was the killer, then the jury committed a Type II error by saying that there is insufficient evidence to support Ha ● Are you familiar with O.J. Simpson’s case? – No Type I error. – Type II error ? • Yes if he was actually guilty Summary - Elements of a Test of Hypothesis ● ● ● ● ● ● ● Null Hypothesis (H0) Alternative or Research Hypothesis (Ha) Test Formula Rejection Region Calculation of the test statistic and conclusion P-value or significance probability Two Types of Errors (can’t have both in one test) ● Note: a test is one-tailed if the rejection region is located on one side of the distribution, otherwise the test is two-tailed. Understanding statistical concepts associated with test of hypothesis ● The smaller the p-value associated with a test of hypothesis, the stronger the support for (a) null hypothesis (b) research hypothesis. » Ans. (b) ● Which elements of a test of hypothesis can be specified prior to collecting data? • (a) Null hypothesis (b) Research hypothesis (c) α (d) all of the above » Ans (d) ● If we do not reject the null hypothesis, we conclude that • (a) There is enough evidence to infer that the alternative hypothesis is true • (b) There is enough evidence to infer that the null hypothesis is true • (c) There is insufficient evidence to support the alternative hypothesis » Ans. (c) ● What is a Type I error? • (a) rejecting the null hypothesis when it is false, (b) rejecting the null hypothesis when it is true, (c) α , (d) both (b) and (c) » Ans. (b) ● What is a Type II error? • (a) rejecting the null hypothesis when it is false • (b) failing to reject the null hypothesis when it is false • (c) rejecting the null hypothesis when it is true (d) β » Ans. (b) Remarks ● Type I Error: Reject null hypothesis H0 when H0 is true • α = p(Type I Error) ● Type II error: Accept H0 when research hypothesis Ha is true • β = p(Type II Error) ● α and β move in opposite direction; Small α results in large β. ● A two-tailed test rejects H0: µ = µ0 in favor of Ha: µ ≠ µ0 for all µ0 that falls outside (1-α)100% confidence interval for µ. ● If you use α= 0.05 for your test, then you are allowed to reject true null hypothesis 5% of the time in repeated application of your test rule. ● If the p-value of a test is 0.04 (say) and you reject H0 then, under your test rule, 4% of the time you would reject true null hypothesis. P-value must be less than α to reject H0; ● A test is called one-tailed or one sided if the rejection region is located on one tail/side of the distribution of your test statistic. Collecting Data and the concept of random error or noise ● Designed Study o Analysts control treatments (independent variables) and assignment of units to treatments and record the value of the response variable ● Observational Study o Analysts observes both treatments and response variable on a random sample of units ● Goal: ● Experimental Noise or Error or random Error o ● Determine whether there is any real difference between treatments and estimate differences The observed results or outcomes of an experiment when repeated/replicated under essentially the same conditions may not be identical. The difference in results that occurs from one repetition to another or from one unit to another identical unit is known as noise or error. Thus an error is unavoidable or unexplained variation in the response variable. Some widely used designs: o o o Independent samples design or completely randomized design Paired design or before-after design or pre-post study Randomized complete block design (paired design is a special case of this design) Simple Linear Regression ● In many situations, the mean of a population is not viewed as a constant, but rather as a variable, dependent on the value of another variable. For example, - mean sale price (y) of houses may depend on square feet (x) of living spaces - mean starting salary (y) of a college graduate may depend on GPA (x) - mean sales amount of ice cream may depend on the temperature (x) ● In simple linear regression, the mean (also called expected value) of a dependent variable y is assumed to be linearly related (straight-line relationship) to a single quantitative independent variable x as ● µy = E(y) =β0 + β1 x ● Understanding intercept (β0) and slope (β1): - For one unit increase in the value of x, the mean of y is increased (decreased) by β1 if β1 is positive (negative). - β0 represents the mean of y when x = 0. Simple Linear Regression Example • Age and height of children • Age = Age in Months; • Height = average height in cm of 161 children of each age (all data values are not available) • Age: 18 19 20 21 22 23 24 25 26 27 28 29 ……. • Height: 76.1 77.0 78.1 78.2 78.8 79.7 79.9 81.1 81.2 81.8 82.8 83.5 ……. Mean Height = a + b*Age Correlation Coefficient • Recall Scatterplots: These plots are used to describe relationship between two quantitative variables. • The correlation coefficient is a quantitative measure of the strength of linear relationship between two variables x and y. • Properties of r: ( -1 ≤ r ≤ 1) • The value of r lies between -1 and +1 including +1 (a perfect positive correlation) and -1 (a perfect negative correlation). • If there is absolutely no correlation present, the value of r is zero. • The closer the number is to 1 or -1, the stronger the correlation, or the stronger the relationship between the two variables. • The closer the number is to 0, the weaker the correlation. • Something that seems to correlate in a positive direction might have a value of 0.8, whereas something with a weak negative correlation might have the value -.3, etc. Multiple Linear Regression ● In simple linear regression, the mean of a population is viewed as a variable, dependent on the value of another variable (mean age (y) of children is assumed to depend on the age (x)) ● In multiple linear regression, the mean (also called expected value) of a dependent variable y is assumed to be related to k(≥ 2) independent variables as ● µy = E(y) = β0 + β1 x1 + β2 x2 + … + βk xk ● Understanding β–parameters - For one unit increase in the value of x1, the mean of y is increased (decreased) by β1 if β1 is positive (negative) for given values of all other variables β0 represents the mean of y when all other variables are set at zero. - ● Example: Insurance agents determine your automobile premium (Y) based on your answer to several questions listed in the application form. Some of the important questions are listed below: • • • • • DMV record (X1) ; Age (X2) ; Driver education course (X3); School/College GPA/Education (X4); Distance and miles of driving (X5) Equipment (Alarm/anti-theft devices, airbags) in your car (X6); Number of cars insured (X7); Place (X8) ; Gender (X9) ; Race (X10) ; Marital Status (X11) ; Credit Score (X12).