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UA MATH 167 - Study Notes

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SOLUTIONSMAT 167: StatisticsTest IIInstructor: Anthony TanbakuchiSpring 2008Name:Computer / Seat Number:No books, notes, or friends. Show your work. You may use the attachedequation sheet, R, and a calculator. No other materials. If you choose to use R,write what you typed on the test or copy and paste your work into a worddocument labeling the question number it corresponds to. When you are donewith the test print out the document. Be sure to save often on a memory stick justin case. Using any other program or having any other documents open on thecomputer will constitute cheating.You have until the end of class to finish the exam, manage your time wisely.If something is unclear quietly come up and ask me.If the question is legitimate I will inform the whole class.Express all final answers to 3 significant digits. Probabilities should be given as adecimal number unless a percent is requested. Circle final answers, ambiguous ormultiple answers will not be accepted. Show steps where appropriate.The exam consists of 13 questions for a total of 84 points on 12 pages.This Exam is being given under the guidelines of our institution’sCode of Academic Ethics. You are expected to respect those guidelines.Points Earned: out of 84 total pointsExam Score:MAT 167: Statistics, Test II SOLUTIONS p. 1 of 12Solution: Spring 2008 results. I made the exam out of 82 points instead of 84.> summary ( s c o re )Min . 1 s t Qu. Median Mean 3 rd Qu. Max.43 .2 9 68 .6 0 81 .1 0 75 .7 5 89 .7 9 95 .1 2> par ( mfrow = c ( 1 , 2 ))> box plot ( s co re , main = ”Boxplot o f exam s c o r e s ”)> h i s t ( s co re , main = ”Histogram o f exam s c o r e s ”)50 60 70 80 90Boxplot of exam scoresHistogram of exam scoresscoreFrequency40 50 60 70 80 900 1 2 3 4 5 61. An experiment consists of randomly sampling 10 students at Pima Community College, record-ing their heights and computing their mean height. If we repeat the experiment over and over(assume simple random samples, no human errors, no bias) we observe that the sample meanheight varies each time.(a) (2 points) What is the name of the error that causes the mean to vary each time?Solution: sampling error(b) (2 points) Explain how it is possible for the sample mean height to vary each time? Whatis going on?Solution: Each time we conduct a random sample of size n from a fixed population,the elements that are selected in the sample will vary. Since the samples vary then wewould expect the sample mean to vary as well. This is the natural variation we wouldexpect to see from sample to sample called sampling error.(c) (2 points) In words, state what the population distribution represents in this experiment.(Be specific.)Solution: A population distribution describes the distribution of population values.In this experiment, the population distribution describes the heights of all students atPima Community College.Instructor: Anthony Tanbakuchi Points earned: / 6 pointsMAT 167: Statistics, Test II SOLUTIONS p. 2 of 12(d) (2 points) In words, state what the sampling distribution represents in this experiment.(Be specific.)Solution: A sampling distribution describes the distribution of statistics for a randomsample of size n. In this experiment, it’s the distribution of all possible sample meansfor n = 10. More specifically, the sampling distribution describes all the possiblesample means that could be obtained from all possible samples of size 10 at PimaCommunity College.2. (2 points) Under what conditions can we approximate a binomial distribution as a normaldistribution?Solution: If the requirements for a binomial distribution are met, it can be approximatedas a normal distribution when : np, nq ≥ 5.3. (2 points) Which distribution (normal, binomial, both, or neither) would be appropriate fordescribing:The distribution for the number of people who wear glasses in a random sample of 20 peoplewhere the probability an individual person wears glasses is 0.56.Solution: Both the normal and the binomial would be appropriate. The binomial dis-tribution is appropriate because there is a fixed number of trials and p does not change(independent), and only two outcomes. The binomial can be approximated as a normalsince np, nq ≥ 5.Instructor: Anthony Tanbakuchi Points earned: / 6 pointsMAT 167: Statistics, Test II SOLUTIONS p. 3 of 124. In regards to ¯x and the Central Limit Theorem:(a) (2 points) What are the two conditions under which the CLT applies?Solution: Either (1) x (the population) has a normal distribution or (2) n > 30.(b) (2 points) If the conditions are met, what type of distribution will ¯x have?Solution: The sampling distribution of ¯x can be described as a normal distribution.5. Let x be a random variable with a normal distribution where µ = 45 and σ = 10.(a) (2 points) Make a meaningful sketch that represents P (50 < x < 55).Solution:20 30 40 50 60 700.00 0.01 0.02 0.03 0.04xf(x)(b) (2 points) Find P (50 < x < 55).Solution: Use normal CDF: P (50 < x < 55) = F (55) −F (50)> pnorm ( 55 , mean = 45 , sd = 10) − pnorm ( 50 , mean = 45 , sd = 10)[ 1 ] 0.1 4988 2 3Instructor: Anthony Tanbakuchi Points earned: / 8 pointsMAT 167: Statistics, Test II SOLUTIONS p. 4 of 126. The following questions regard hypothesis testing in general.(a) (2 points) When we conduct a hypothesis test, we assume something is true and calculatethe probability of observing the sample data under this assumption. What do we assumeis true?Solution: We assume the null hypothesis H0is true.(b) (2 points) If you are using a hypothesis test to make a decision where the effect of a TypeI error may negatively effect human lives, should you increase or decrease α?Solution: You should decrease α to reduce the probability of making a Type I error.(c) (2 points) If you failed to reject H0but H0is false. What type of error has occurred?(Type I or Type II)Solution: Type II(d) (2 points) What variable do we use to represent a Type II error with?Solution: β(e) (2 points) Two studies were conducted, study A had a power of 0.9 and study B had apower of 0.10. Which study would be more likely to support a true alternative hypothesis?Solution: Study A because it had a power of 0.90. There is a 90% chance of support-ing a true alternative hypothesis in this study.7. (2 points) Describe what a normal Q-Q plot should look like for a set of data that has a normaldistribution.Solution: It should be a scatter plot of data points that follow a straight line with noobvious pattern. Below wold


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UA MATH 167 - Study Notes

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