Hypothesis Testing and Confidence Intervals in Statistics - Prof. Samiran Sinha, Exams of Statistics

Download Hypothesis Testing and Confidence Intervals in Statistics - Prof. Samiran Sinha and more Exams Statistics in PDF only on Docsity! Final/Exam #3 Form B - Statistics 211 (Fall 1999) • This test consists of nine numbered pages. Make sure you have all 9 pages. It is your respon- sibility to inform me if a page is missing!!! • You have at least 100 minutes to complete this test. • Make sure you fill out and “bubble in” the following items on the scantron sheet: – Last Name, First Name, MI. – Dept (STAT), Course No (211), Section (504). – Social Security Number (enter your Student ID). – Test Form (A or B). – Enter the exam number, sign and date the form. You will loose credit if you fail to fill out your scantron form correctly. • You may use crib sheets 1, 2 and 3 and all appropriate tables and a calculator. • If I provide partial results—assume they are correct and use them even if they are not. • If there is no correct answer or if multiple answers are correct, select the best answer. • There is no penalty to wrong answers. . . so guess if you do not know an answer. • It is your responsibility to look at the overhead or blackboard about every 15 minutes and to incorporate any relevant information into your test. • You can keep this exam! 1 Fill out your scantron sheet properly. See the header page of this exam for instructions. Chapter 8 [1] A type I error is: a) Reject H0 when H0 is false. b) Reject H0 when H0 is true. c) Fail to reject H0 when H0 is false. d) Fail to reject H0 when H0 is true. e) not relevant to a test. [2] A statistical hypothesis is a claim about the value of one or more. . . a) statistics. b) parameter estimates. c) tests. d) confidence intervals. e) population parameters. [3] For a test α = .05 and β = .10. What is the probability we reject H0? a) .05 b) .1 c) .9 d) .95 e) We cannot know this without the data! [4] The desired percentage of SiO2 in a certain type of aluminous cement is 5.5. We want to test whether the true average percentage is 5.5 for a particular production facility. Suppose the percentage of SiO2 in a sample is normally distributed with σ = .3. What sample size n is required to satisfy α = .01 and β(5.6) = .01? a) n = 15 b) n = 20 c) n = 194 d) n = 217 e) n = 721 [5] The one sample test for µ when data is Normal and σ is known is: Z = x̄−µ0 s/ √ n . When our sample size is large enough (say n ≥ 30) we say it is no longer necessary that the data is Normal. Why? a) When we have large samples, x̄ is a better estimate if µ! b) As the sample size increases, the variance of x̄ decreases! c) When we have a large sample size, s is an accurate estimate of σ! d) When n ≥ 30 the CLT no longer holds allowing us to ignore with the Normality assumption. e) The CLT guarantees that x̄ is Normal, even if the data is not. [6] Suppose a manufacturer claims their VHS tapes can hold 120 minutes of programming at SP mode. You believe they are shorter, so you test: H0 : µ = 120 HA : µ < 120 In this example a type II error would be: a) You claim the tapes are shorter than 120 minutes when in reality they are not. 2 are interested if force after impact is greater for advanced players than it is for intermediate players. 1) Advanced n = 6 x̄ = 40.3 s = 11.3 2) Intermediate n = 8 x̄ = 21.4 s = 8.3 The appropriate hypothesis for the above test is: a) H0 : µ1 = µ2 vs Ha : µ1 − µ2 < 0 b) H0 : µ1 = µ2 vs Ha : µ1 − µ2 > 0 c) H0 : µ1 = µ2 vs Ha : µ1 − µ2 6= 0 d) H0 : µd = 0 vs Ha : µd < 0 where µd = µ1 − µ2 e) H0 : µd = 0 vs Ha : µd > 0 where µd = µ1 − µ2 [16] The appropriate test for the above “Tennis elbow” problem is: a) The pooled t-test. b) The Smith-Satterthwaite test. c) The paired t-test. d) Test for two population proportions. e) Test for two population variances. [17] At the beginning of the semester we talked about the Salk Polio vaccine. In a large 1954 experiment 401,974 children where vaccinated. 201,229 where given the vaccine and 200,745 where given a placebo. 110 of the children who received a placebo got polio and 33 of those given the vaccine got polio. Was the vaccine effective? H0 : p1 − p2 = 0 HA : p1 − p2 < 0 Vaccine: m = 201229 x = 33 p̂1 = .000164 Placebo: n = 200745 y = 110 p̂2 = .000548 The test statistic is Z = −6.46. What is the p-value of this test? What is the conclusion? a) less than .0001 — the vaccine is not effective! b) less than .0001 — the vaccine is effective! c) greater than .9999 — the vaccine is not effective d) p-value cannot be calculated, but the vaccine is effective. e) p-value=.0228 — the vaccine is effective! Chapter 10/12 [18] A builder is interested in whether location has an effect on the selling price of a new three- bedroom home in the 2000 square foot range. To keep matters simple, let suppose there are only three areas in which new homes are being built and thus only three treatment levels. The builder samples the price of 3 houses for each area (treatment level) and conducts the test: H0 : µ1 = µ2 = µ3 Ha : notH0 Here is the partial ANOVA. What is the f statistic for this test? Source DF Sum of Squares Mean Square f Treatment 235.67 Error — Total 469.67 — — 5 a) f = 1.68 b) f = 0.33 c) f = 4.03 d) f = 3.02 e) f = 234.00 [19] In the above “builder example” the three different average house prices are (for the three different areas) are: x̄1 = 90 x̄2 = 100 x̄3 = 105 We conduct a Tukey multiple comparison test where w = 7. Which conclusion below is the correct one? a) Areas 1 & 2 are the same, but they are significantly different from area 3. b) Areas 1 & 3 are the same, but they are significantly different from area 2. c) Areas 2 & 3 are the same, but they are significantly different from area 1. d) Areas 1 & 2 & 3 are all significantly different from each other. e) None of the areas 1 & 2 & 3 are significantly different from each other. Chapter 3-6 (selected sections) [20] Two desirable properties of a point estimate θ̂ are: a) Small bias and large variance. b) Large bias and large variance. c) Small bias and small variance. d) Large bias and small variance. e) Only the bias matters! Not the variance. [21] If GPA at Texas A&M is distributed X ∼ N(µ = 2, σ = 0.8) then what GPA must you have to be in the 75th percentile (select the closest answer)? a) 0.675 b) 1.28 c) 2.54 d) 3.00 e) 3.75 [22] The average number of hurricanes that hit the west cost is 0.5 per year. What is the probability more than 3 years pass without a hurricane? a) P (X ≥ 3) where X ∼ Poisson(λ = 0.5). b) P (X ≥ 3) where X ∼ Gamma(α = 1.5, β = 0.5). c) P (X ≥ 3) where X ∼ Gamma(α = 1.0, β = 1.0). d) P (X ≥ 3) where X ∼ Exponential(λ = 0.5). e) P (X ≥ 3) where X ∼ N(µ = 0.5, σ = 1). [23] If two variables X and Y are positively correlated this might mean: a) X causes Y . b) X and Y cause each other. c) X and Y could be caused by a third lurking variable. d) X and Y are related by chance. e) All of the above. [24] The Central Limit Theorem is important because it: 6 a) tells us the distribution of the data even if the distribution of the average is not known. b) tells us the distribution of the data even if the mean and variance of the data are not known. c) tells us the distribution of the average even if the distribution of the data is not known. d) allows us to solve probability questions about X (an individual data point). e) says that X̄ has a binomial distribution if n is large. [25] The average number of rabits per acre in a 7 acre forest is estimated to be 4. Find the probability that 2 or fewer rabits are found on any given acre. a) P (X ≤ 2) where X ∼ Negative Binomial(r = 4, p = 27 ). b) P (X ≤ 1) where X ∼ Binomial(n = 4, p = 27 ). c) .238 d) .146 e) .092 [26] When a baseball player hits .300, he gets a hit 30% of the times at bat. Typical major leaguers bat about 500 times a season and hit about .260. Assuming a hitter’s successive tries are independent, what is the probability that a “typical” .260 player bats .200 or better? a) P (X ≥ .20) when X ∼ Binomial(n = 500, p = .26). b) P (X ≥ 100) when X ∼ Binomial(n = 500, p = .26). c) P (X ≥ .20) when X ∼ Binomial(n = 500, p = .20). d) P (X ≥ 100) when X ∼ Binomial(n = 500, p = .20). e) P (X ≥ 200) when X ∼ Negative Binomial(n = 500, p = .30). [27] The probability a student passes a class is 85%. The average class size is 90 students and the avergae GPA is 2.0. Assuming successive attempts are independent, and assuming 20% of the students hate the class, we want to know how many times a student needs to take class before she passes. The relevant distribution to answer this question is? a) X ∼ Negative Binomial(r = 2, p = .85). b) X ∼ Negative Binomial(r = 2, p = .20). c) X ∼ Geometric(p = .85). d) X ∼ Geometric(p = .20). e) X ∼ Binomial(n = 90, p = .85). Chapter 7 [28] You desire to construct a confidence interval of your estimate of average price of a six pack of Diet Coke in Bryan/College Station. How many store prices must you sample so that your 90% confidence interval is equal to or smaller than ±10 cents? You estimate that σ is 30 cents. a) 24 b) 25 c) 35 d) 97 e) 98 [29] Which of the following will not affect the width of the confidence interval? a) Alpha. b) Standard Deviation. c) Sample size. d) All of the above will affect the width. e) None of the above will affect the width. 7