10 Solved Questions on Elementary Statistics Inference - Final Exam | MTHSC 203, Exams of Mathematics

Download 10 Solved Questions on Elementary Statistics Inference - Final Exam | MTHSC 203 and more Exams Mathematics in PDF only on Docsity! Final Exam Covering Chapters 1 - 10 and 12.2 Name: Instructions: Answer free response with as much work as possible. I can’t give partial credit without work, and the final answer is worth very little. The exam is out of 250 points so I don’t have to use fractions. (1) Chapter 1 (a 2pts) Give an example of a sample statistic: Sample mean, sample variance, sample median, etc (b 2pts) Give an example of a population parameter: mean, variance, median, etc (c 2pts) Give an example of qualitative data: answers like “very much”, “a little”, etc (d 2pts) Give an example of quantitative data: measurements (numeric), heights, widths, weights, etc (e 2pts) Give an example of discrete data: counts, integers, shoe sizes, etc (f 2pts) Give an example of continuous data: real numbers, time, distance, weight, etc (2) Chapter 2. Using this list of data: 6, 5, 4, 5, 6, 5, 4, 5, 6, 4, 4, 2. (a 10pts) What is the frequency distribution, relative frequency distribution, and cumulative frequencies for the above data? frequency relative frequency cumulative frequency 2 1 1/12 1 3 0 0 1 4 4 4/12 5 5 4 4/12 9 6 3 3/12 12 7 0 0 12 (b 5pts) What are the key parts to drawing a frequency histogram (either talk through it OR do it for the list above)? It’s a bar graph with no spaces between the bars. The bar heights correspond to the number of times (frequency) the value shows up (3) Chapter 3 (a 2pts) What are two statistics that measure center? sample mean, sample median (b 2pts) What is the sample mean of this list of data 4, 3, 3, 7, 5, 8? 4+3+3+7+5+86 = 5 (c 2pts) What are two statistics that measure the spread of data? sample variance, sample standard deviation, quartiles (d 4pts) What is the sample standard deviation of this list of data 4, 3, 3, 7, 5, 8?√ (4− 5)2 + (3− 5)2 + (3− 5)2 + · · ·+ (8− 5)2 5 = √ 4.4 = 2.098 (e 3pts) If I told you that µ = 15 and σ = 5, what is the z-score for x = 25? Z = x− µ σ = 25− 15 5 = 2 1 (4) Chapter 4: Suppose we poll random people and X = 1 when they say yes and X = 0 when they say no. (a 2pts) What is the sample space for just one X? Zero or one, ie {0, 1} (b 2pts) What is the sample space for p̂? Multiples of 1/n between zero and one, ie {0, 1/n, 2/n, 3/n, . . . , n/n} (c 4pts) What is the difference between an outcome and an event? If you prefer, use the above setup and reason through example. An outcome is a smallest possible result possible. An event is a collection of (one or more) outcomes. (c 4pts) What are the 3 rules of a probability? • 0 ≤ P (A) ≤ 1 for all events A • P (∅) = 0 and P (S) = 1 where S is the sample space • Whenever A and B are disjoint events, P (A ∩B) = 0 (d 4pts) Using the population below: if we pick someone randomly, what is P ( art major | left handed )? right handed left handed art major 192 8 science major 195 105 P (art|left) = P (art and left) P (left) = 8/500 113/500 = 8 113 Notice, there’s 500 total listed in the table (If you were wondering where 500 came from) (e 4pts) Using the population above, if we select 5 individuals (no repeats), what is the probability that at least one of them is left handed? I recommend answering this without trying to use your calculator, it’ll probably give an error. (Hint: order doesn’t matter) P (at least one) = 1− P (none) = 1− ( 387 5 )( 500 5 ) (5) Chapter 5. (a 3pts) Is a normal random variable discrete or continuous? continuous (b 3pts) Is a Bernoulli random variable discrete or continous? discrete (c 6pts) Suppose you poll n = 10 individuals each with probability of success p = .25. Let X be a random variable that counts the number of successes. What is the name and parameters of the distribution of X? X is Binomial(n=10,p=0.25) (d 2pts) Using the addition rule and the setup for part (c), what is P (X < 2) in terms of P (X = 0) and P (X = 1)? P (X < 2) = P (X = 0) + P (X = 1) since {X = 0} and {X = 1} are disjoint events (e 6pts) Using the setup for part (c), calculate P (X < 2). P (X = 0) = ( 5 0 ) p0(1− p)5 P (X = 1) = ( 5 1 ) p1(1− p)4 and P (X < 2) = P (X = 0) + P (X = 1) 2 (d 10pts) Suppose we’re testing Ho : σ = 2 vs H1 : σ 6= 2. With n = 23 observations, you observe an X = 2.31 and S2 = 4.3. Using the confidence interval method, and working at α = 0.05, what conclusion can you make? Careful, this is a claim about σ. With 22 degrees of freedom and 5% in each tail, we get critical points of χ2L = 10.982 and χ 2 R = 36.781. Then a 95% CI for σ is:√ (n− 1)S2 χ2R ≤ σ ≤ √ (n− 1)S2 χ2L or √ 22 · 4.3 36.781 ≤ σ ≤ √ 22 · 4.3 10.982 or (1.6, 2.9) Therefore, we can’t reject the claim since 2 is inside the interval. (e 5pts) Suppose we’re testing Ho : p = .5 vs H1 : p < .5. With n = 100 observations we got X = 35. What is the p-value of this observation? Z = p̂− p√ p(1−p) n = .35− .5√ 0.5(1−0.5) 100 = −3 is our test statistic. The p-value then is p-value=P (Z < −3) = 0.0013 (9) Chapter 9. Whenever asked for the test statistic, be sure to tell which distribution it follows and how many degrees of freedom (if applicable). (a 5pts) When comparing two population proportions, which test statistic applies? Z = (p̂1 − p̂2)− (p1 − p2)√ p(1−p) n1 + p(1−p)n2 (b 5pts) When comparing two population means with σ1 and σ2 given, which test statistic applies? Z = (X1 −X2)− (µ1 − µ2)√ σ21 n1 + σ 2 2 n2 (c 5pts) Why would you pool the sample variances together? Why shouldn’t you in practice? It increases (just about combines) your degrees of freedom (this is good). The problem is that in practice, there’s no reason to assume that σ1 = σ2. (d 10pts) Suppose you do an experiment with two independent populations to get n1 = 100, X1 = 2.6, S1 = 1.2, n2 = 150, X2 = 2.2, and S2 = 1.5. Using the CI method, test the claim that Ho : µ1 − µ2 = 0 vs H1 : µ1 − µ2 6= 0 with α = 0.05. A 95% CI for µ1 − µ2 is: X1 −X2 ± tα/2 √ S21 n1 + S22 n2 = 0.4± 1.984 ∗ 0.171 = (0.06, 0.74) Since Ho is not inside the interval, we reject the claim that µ1 equals µ2. (e 10pts) Using the table of matched pairs below and α = 0.05, test H0 : µ1 = µ2 vs H1 : µ1 6= µ2. person1 person2 person3 measurement 1 1 2 2 measurement 2 0 2 3 1 0 -1 Using the differences in the 3rd row, find X and S2 X to test the claim µd = 0. You’ll find that you can’t reject. 5 (f 5pts) Suppose you want to test the claim that the variances of your two independent samples are the same. How do you distinguish between the two populations? What is the test statistic and how must the original population(s) be distributed to use this test statistic? Well, we define the first population so that S21 > S 2 2 . Then our test statistic is: F = S21 S22 with degrees of freedom: n1 − 1 n2 − 2 (10) Chapter 10. (a 3pts) Which of the data plots from the earlier chapters is useful for bivariate data? Scatterplot (b 5pts) Given the least squares regression line ŷ = 2 + 0.67x and (x, y) = (3, 4), what is your best prediction of y given x = 6 when ρ 6= 0? Pluggin in x, we get ŷ = 6. (c 5pts) Given the least squares regression line ŷ = 2 + 0.67x and (x, y) = (3, 4), what is your best prediction of y given x = 6 when ρ = 0? Use Y = 4 (d 10pts) Given that Sxy = −16, Sx = 4, Sy = 5, x = 3.4, y = 5.2, and n = 22. What is your best point estimate for ρ and can you conclude (think hypothesis test!) that ρ 6= 0? Our best point estimate for ρ is r = SxySxSy = −16/20 = −0.8. With n− 2 = 20 deg of freeom, we have the following test statistic: t = r√ 1−r2 n−2 = −0.8√ 1−0.82 20 = −5.96 And we would reject for any reasonable choice of α. (e 10pts) Using the same setup for part (d), what is the least squares regression line? We’re finding b1 and bo for ŷ = bo + b1x. b1 = r Sy Sx = Sxy S2x = −1 bo = Y − b1 ·X = 5.2 + 3.4 = 8.6 Since the line goes through the centroid. Finally, our line is ŷ = 8.6− x 6 (f 2pts) You saw in part (b) that you can make predictions when there’s a linear correlation. What is a more “certain” form of estimation than estimating with a single point? (I only want 2 words) Prediction intervals! (11 10pts) Suppose you want to test the claim that of the 4 populations you measure all their means are the same. You’re given the following table of data (with statistics at the bottom). Test the following claim: H0 : µ1 = µ2 = µ3 = µ4 vs H1 : nope. population1 population2 population3 population4 observation 1 44 45 57 55 observation 2 45 53 57 47 observation 3 39 49 46 59 observation 4 58 51 54 49 observation 5 44 46 51 42 x 46 48.8 53 50.4 S 7.11 3.35 4.64 6.69 Finding the sample variance of the 2nd to last row, we get S2 X = 8.60. Finding the sample mean of the last row (squaring the S’s to get S2’s) gives XS2 = 32.01. Now our test statistic is: F = 5 · 8.60 32.01 = 1.34 with df= k − 1 k(n− 1) = 3 16 With a critical point of 4.08, the test statistic is well to the left. Therefore we fail to reject with α = 0.05. 7