STAT 210 1nd EditionExam # 4 Study Guide Lectures: 19-22Lecture 19 (October 6)- To describe a distribution, must describe the center, spread, shape, and unusual features. - 2 types of distributions: normal and t-distributions- Normal curves are symmetric and bell shaped- Normal curve data follows a normal distribution, which is an example of a continuous distribution- X ~ N (µ , σ)o “X is distributed normal with mean µ and standard deviation σ.” X= name of a variable x= value of the variable- Properties of Normal Distributiono Normal curve is bell shapedo Peal of curve is the population mean µo Normal curve is symmetric about µo Center and spread completely specified by specifying the values of population mean µ and the population standard deviation σo Total area under the normal curve is 1 (or 100%)o 68-95-99.7% rule 68% of the measurements fall within one standard deviation σ of the mean µ - µ - σ- µ + σ 95% of the measurements fall within two standard deviations (2σ) of the mean µ- µ - 2σ- µ + 2σ 99.7% if the measurements fall within three standard deviations (3σ) of the mean µ- µ - 3σ- µ + 3σ- There are two types of problems for this section:1. Given values of X, find probability (or area/proportion/percentage).i. Find P(X = x)ii. Find P(X < x) or P(X < x)iii. Find P(X > x) or P(X > x)iv. Find P(x1 < X < x2) or P(x1 < X < x2)2. Given a probability (or area/proportion/percentage), find the value of X.i. Find the value x so the probability of being less than/less than andequal to the value is as specifiedii. Find the value x so the probability of being greater than/greater than and equal to the value is as specifiediii. Find x1 and x2 so the probability of being between the values is as specified- Because normal distribution is continuous, for any value x the probability that a normal variable X equals the value of x is 0.o P(X= x)= 0- Standard Normal Distributiono Denoted as Zo Population mean µ = 0 (center)o Standard deviation σ = 1 (spread)o Symmetric bell curveo No unusual featureso Z ~ N (0,1)- To read normal Z table to find probabilities, read value of z down the left and across the top, then read the probability from the body of the table.- For equal to problems, since standard normal distribution is a normal distribution for any value z, then P(Z=z)=0.- For less than problems, look at the normal Z table of probabilities. Look up the number before the decimal and the first number after the decimal in the left column, then look up the second number after the decimal in the top row. - For greater than problems: P(Z > z)= 1-P(Z < z)- For between problems, convert into two less than problems and subtract the smaller probability from the larger. Lecture 20 (October 8)- Probability is given and want to find corresponding value of Z. Three such problems:1. Less than problemsa. Want to find value z so that the probability of being less than z is as specified.b. Draw a normal curve. Mark the info given in problem.c. Find the less than probability in body of normal table, read across and up, determine the z value.2. Greater than problemsa. Draw a normal curve. Mark the info given in problem.b. Convert from greater than to less than (subtract the given probability from 1)c. Find the less than probability in body of normal table, read across and up, determine z value.3. Between problemsa. Subtract given value from 1b. Because symmetric distribution, divide that number by 2 to get probability of z1 being less than that value and z2 being greaterc. Find value of z1 by looking at tabled. Find value of z2 by converting to less than problem- Standard normal (Z) distribution requires mean is 0 and standard deviation is 1- If X is distributed with mean not equal to 0 and/or standard deviation not equal to 1, must do Z-Score Transformationo Z = value-mean/standard deviationLecture 21 (October 10)- Z-Score Transformationo Find z value that satisfies probabilityo Calculate x X= mean + Z(standard deviation)Lecture 22 (October 13)- When population standard deviation unknown, use t-distributions instead of standard normal distribution- Z-distribution and t-distribution have same symmetric bell-shape, same mean (0), & no unusual features.- T-distribution standard deviation greater than/equal to 1. Depends on degrees of freedom (df); as df increases the t-distribution gets closer to z-distribution- Make sure to understand how to read t-distribution tables- Sampling distribution is the distribution of values taken by statistic in large number of simple random samples of same size from same