Math 1040 Final Exam Form A Introduction to Statistics Spring Semester Name Section Instructor

Transcription

1 Math 1040 Final Exam Form A Introduction to Statistics Spring Semester 2015 Name Section Instructor Time Limit: 120 minutes Any calculator is okay. Necessary tables and formulas are attached to the exam. All problems are weighted equally. Computers, cell phones, and hand-held devices other than calculators are not allowed. Students may not bring notes, formulas, or tables into the exam. This exam has two parts Part 1 - Ten multiple choice questions Part 2 - Ten open-ended questions INSTRUCTIONS PART I: Questions 1-10, Multiple choice. Answer all TEN questions and circle the correct answer. It is not necessary to show work. There will be no partial credit awarded on this part of the exam. 1)Does regular exercise reduce the risk of a heart attack? Here are 2 ways to study the question: Study 1: A researcher finds 2000 men who are over 40 who exercise regularly and have not had a heart attack. She matches each with a similar man who does not exercise regularly, and follows both groups for 5 years to see what percent had a heart attack. Study 2: A researcher finds 4000 men aged over 40 who have not had heart attacks and are willing to participate. She randomly assigns 2000 of them to a regular program of supervised exercise. The other 2000 continue with their usual habits. She follows them for 5 years to see what percent had a heart attack. Are these experiments or observational studies? A) Both Study 1 and Study 2 are observational studies. B)Both Study 1 and Study 2 are experiments. C) Study 1 is an experiment, Study 2 is an observational study. D) Study 1 is an observational study, Study 2 is an experiment. Form A page 1

2 2)Sara and Jerry took a math exam. Sara's percentile score on the exam was 35; Jerry's percentile score on the same test was 70. We know that A) Sara scored better than 35 of her classmates. B)Sara correctly answered half as many items as Jerry did. C) They both scored better than average on the math exam. D) Jerry correctly answered more items than Sara did. 3) Shawn and Maddie purchase a foreclosed property and spend additional funds fixing it up. They feel that they can resell the property for $120,000 with a probability 0.15, $100,000 with probability 0.45, $80,000 with probability 0.25, and $60,000 with probability Compute the expected selling price (expected value) of the property. A) $88,000 B) $90,000 C) $92,000 D) $96,000 4) Which of the following statements is NOT TRUE according to the Central Limit Theorem? A) An increase in sample size from n=16 to n=25 will produce a sampling distribution with a smaller standard deviation. B)The mean of a sampling distribution of sample means is equal to the population mean divided by the square root of the sample size. C) The mean of the sampling distribution of sample means for samples of size n=15 will be the same as the mean of the sampling distribution for samples of size n=100. D) The larger the sample size, the more the sampling distribution of sample means will resemble a normal distribution. Form A page 2

3 5)A janitor at a large office building believes that his supply of light bulbs has a defect rate that is different than the defect rate stated by the manufacturer. The janitor's null hypothesis is that the supply of light bulbs has a defect rate of p = 0.09 (the light bulb manufacturer's stated defect rate). Suppose we perform a hypothesis test with a significance level of Symbolically, the null and alternative hypothesis are as follows: H 0 : p = 0.09 and H 1 : p > Choose the statement that best describes the significance level in the context of the hypothesis test. A) The significance level of 0.01 is the probability of concluding that the defect rate is equal to 0.09 when in fact it is greater than B)The significance level of 0.01 is the z-statistic that we will use to compare the observed outcome to the null hypothesis. C) The significance level of 0.01 is the defect rate we believe is the true defect rate. D) The significance level of 0.01 is the probability of concluding that the defect rate is different than 0.09 when in fact the defect rate is equal to ) A skeptical paranormal researcher claims that the proportion of Americans that have seen a UFO, is less than 2 in every ten thousand. If a hypothesis test of the claim is performed, how should you interpret a decision that fails to reject the null hypothesis? A) There is sufficient evidence to support the claim that the true proportion is less than 2 in ten thousand. B)There is not sufficient evidence to support the claim that the true proportion is less than 2 in ten thousand. C) There is sufficient evidence to support the claim that the true proportion is greater than 2 in ten thousand. D) There is not sufficient evidence to support the claim that the true proportion is greater than 2 in ten thousand. Form A page 3

4 7)This table records the number of deaths for the top thirteen worst U.S. tornados since A histogram showing the distribution is also included. For the tornado data, choose the most appropriate measure of center, then calculate the value rounded to the nearest tenth. A) Mean; B) Median; C) Median; D) Mean; ) The probability that an individual is left-handed is Professor Smart's statistics class has 40 students. What is the probability of finding exactly five left-handers in the class? A) B) C) D) 0.11 Form A page 4

5 9)A random sample of college students was asked to respond to a survey about how they spend their free time on weekends. One question, summarized in the table below, asked each respondent to choose the one activity that they are most likely to participate in on a Saturday morning. The activity choices were homework, housework, outside employment, recreation, or other. What is the probability that a randomly chosen survey respondent is male OR chose "recreation" as their most likely activity on Saturday mornings? A) B) C) D) None of these 10) A sociologist develops a test to measure attitudes about public transportation, and 27 randomly selected subjects are given the test. Their mean score is 76.2 and their standard deviation is Construct the 95% confidence interval for the standard deviation, σ, of the scores of all subjects. Assume that the population has a normal distribution. Round the confidence interval limits to the nearest tenth. A) 17.2 < σ < 27.2 B)16.6 < σ < 28.6 C) 16.9 < σ < 29.3 D) 17.5 < σ < 27.8 Form A page 5

6 INSTRUCTIONS PART II: Questions 11-20, Open-Ended. Answer all TEN questions carefully and completely, showing your work and clearly indicating your answer. Partial credit may be awarded for relavent work. 11) Julie teaches a small section of Introductory Statistics. One evening, she asked each of her 17 students to report the number of hours they spend each week (on average) working on their homework assignments. These are the responses given by these 17 students: 0, 1, 3, 4, 5, 6, 6, 6, 7, 9, 10, 11, 11, 13, 15, 18, 23 Create a frequency histogram for this data. Use 6 classes, with 0 as the lower class limit of the first class. Label your histogram with lower class limits. Be sure to label axes appropriately and to title your graph. 12) The distribution of heights of adult men is approximately normal with mean 69.0 inches and standard deviation 2.5 inches. What height marks the 90th percentile? Report the height to the nearest tenth of an inch. Include in your solution a clearly labeled sketch of the appropriate graph. Form A page 6

7 13)A group of 19 randomly selected employees has a mean age of 22.4 years with a standard deviation of 3.8 years. Construct a 99% confidence interval for the population mean, μ. Assume the population has a normal distribution. Round to the nearest tenth. 14) The total medal counts for the 2010 Winter Olympics are as follows (in descending order of medal count): United States 37 Germany 30 Canada 26 Norway 23 Austria 16 Russia 15 South Korea 14 Sweden 11 China 11 France 11 Switzerland 9 Netherlands 8 Czech Republic 6 Poland 6 Italy 5 Japan 5 Finland 5 Australia 3 Slovakia 3 Belarus 3 Slovenia 3 Croatia 3 Latvia 2 Great Britain 1 Kazakhstan 1 Estonia 1 Report the following measures for this dataset, rounding to the nearest hundredth. Mean Mode Median Range Standard Deviation 15) Using the 2010 medal counts data, draw a boxplot representing the data. Report the five-number summary (including the names or symbols of the reported statistics). Be careful to appropriately scale and label the boxplot. Form A page 7

8 16)According to Harper's magazine, the hours spent in front of a television per year by children can be modelled by a normal distribution with a mean equal to 1500 hours and a standard deviation equal to 250 hours. What is the probability that a sample of 49 children will have a mean of less than 1450 hours of TV per year? Include in your solution a clearly labeled sketch of the appropriate graph, shading the area that corresponds to the probability. 17) A television sports commentator wants to estimate the proportion of Americans who watched the 2010 NBA All-Star Game on TV. What sample size should be used if he wants to be within 3 percentage points with 95% confidence if he does not use any prior estimates? Form A page 8

9 18) Suppose a consumer product researcher wanted to find out whether a printer ink cartridge lasted longer than the manufacturer's claim that their ink cartridges could print 400 pages. The researcher tested 40 ink cartridges and recorded the number of pages that were printed before the ink started to fade. At the α=0.05 level, test the hypothesis that the ink cartridges lasted for more than 400 pages, on average. For the 40 cartridges tested, the researcher found a sample mean of 415 pages and a sample standard deviaiton of 30 pages. Round values to the nearest thousandth as needed. Null Hypothesis: Alternative Hypothesis: Test Statistic: P-value (or critical value(s): Conclusion about the Null Hypothesis: Conclusion addressing the original claim: Form A page 9

10 19) The data below are the gestation periods, in months, of randomly selected animals and their corresponding life spans, in years. Gestation, x Life span, y Determine if there is significant linear correlation between Gestation Period and Life Span. Include the value of the correlation coefficient and the appropriate critical value for α=0.05 in your response. 20)a. Use the Gestation Period and Life Span data to find the equation of the least-squares regression line. Round the coefficients to three decimal places, if necessary. b. Use a complete sentence to interpret the meaning, in the context of this data set, of the slope coefficient of the regression line. Form A page 10

11 Descriptive Statistics Probability Statistics Formulas and Tables x = x n x = (f x) f mean s = (x x )2 n 1 s = (x i x ) 2 f i f i 1 variance = s 2 approximate mean from a frequency table standard deviation approx. std. dev. from a frequency table Interquartile Range: IQR = Q 3 Q 1 Lower fence = Q 1 1.5(IQR) Upper fence = Q (IQR) General Addition Rule P(A or B) = P(A) + P(B) P(A and B) multiplication rule for independent events P(A and B) = P(A) P(B) multiplication rule for dependent events P(A and B) = P(A) P(B A) complement rule P(A ) = 1 P(A) n P r = n! Permutations (no elements alike) (n r)! n! n 1!n 2! n k! Permutations (n 1 alike, ) n C r = n! Combinations (n r)!r! Probability Distributions mean (expected value) of a discrete random variable μ = [x P(x)] standard deviation of a discrete random variable σ = [x 2 P(x)] μ 2 P(x) = C x n p x (1 p) n x Binomial probability μ = n p mean for a Binomial distribution σ = n p (1 p) std. dev. for a Binomial distribution Estimating a Population Parameter Proportion: pˆ E p pˆ E where E z z pˆ 0.25 n sample size, p unknown 2 E 2 z ˆ 2 1 ˆ p p 1 pˆ n 2 E sample size, p known Mean: s x E x E where E t 2 n z 2 n sample size E Standard Deviation: n s n s R L n Normal Distribution and Sampling Distributions z = x μ σ standard normal score mean of the sampling distribution of x μ x = μ std. dev. of the sampling distribution of x (std. error) σ x = σ n μ p = p mean of the sampling distribution of p σ p = p(1 p) std. dev. of the sampling distribution of p n Hypothesis Testing z pˆ p p 1 p proportion one population n x t mean one population s n Linear Correlation and Regression r = linear correlation coefficient r = ( x i x sx )(y i y sy ) n 1 y = b 0 + b 1 x estimated eqn. of linear regression line R 2 = r 2 coefficient of variation Residual = y y

12 B-7 Formulas and Tables by Mario F. Triola Copyright 2014 Pearson Education, Inc. Ch. 13: Nonparametric Tests (x + 0.5) - (n>2) z = 1n 2 z = T - n (n + 1)>4 n (n + 1)(2n + 1) B 24 z = R -m R s R = H = r s = 1 - Sign test for n 7 25 R- n 1(n 1 + n 2 + 1) 2 B n 1 n 2 (n 1 + n 2 + 1) 12 Wilcoxon rank-sum (two independent samples) 12 N(N + 1) ar2 1 + R R2 k b - 3(N + 1) n 1 n 2 n k Kruskal-Wallis (chi-square df = k - 1) 6Σd 2 n(n 2-1) acritical values for n 7 30: z = G -m G s G = Ch. 14: Control Charts R chart: Plot sample ranges UCL: D 4 R Centerline: R LCL: D 3 R x chart: Plot sample means UCL: x + A 2 R Centerline: x LCL: x - A 2 R Rank correlation p chart: Plot sample proportions p q UCL: p + 3 B n Centerline: p p q LCL: p - 3 B n Wilcoxon signed ranks (matched pairs and n 7 30) { z 1n - 1 b G -a 2n 1n 2 n 1 + n 2 + 1b (2n 1 n 2 )(2n 1 n 2 - n 1 - n 2 ) B (n 1 + n 2 ) 2 (n 1 + n 2-1) Runs test for n 7 20 Table A-6 Critical Values of the Pearson Correlation Coefficient r n a = a = NOTE: To test H 0 : r = 0 against H 1 : r 0, reject H 0 if the absolute value of r is greater than the critical value in the table. Control Chart Constants Subgroup Size n D 3 D 4 A !! " # $ % # & ' ' ( ) * +, & & 0 ( -. - (. ( / 1 2

13 B-1 Table A-2 Standard Normal (z) Distribution: Cumulative Area from the LEFT z and lower * * NOTE: For values of z below -3.49, use for the area. *Use these common values that result from interpolation: z score NEGATIVE z Scores Area z _Barrelfold_pp01-08.indd 1 9/26/12 9:52 AM

14 B-2 POSITIVE z Scores 0 z Table A-2 (continued) Cumulative Area from the LEFT z * * and up.9999 NOTE: For values of z above 3.49, use for the area. *Use these common values that result from interpolation: z score Area Common Critical Values Confidence Critical Level Value _Barrelfold_pp01-08.indd 2 9/26/12 9:52 AM

15 B-3 Table A-3 t Distribution: Critical t Values Degrees of Freedom Area in One Tail Area in Two Tails Large _Barrelfold_pp01-08.indd 3 9/26/12 9:52 AM

16 B-4 Table A-4 Chi-Square (x 2 ) Distribution Area to the Right of the Critical Value Degrees of Freedom Source: From Donald B. Owen, Handbook of Statistical Tables. Formulas and Tables by Mario F. Triola Copyright 2014 Pearson Education, Inc. Degrees of Freedom n - 1 Confidence Interval or Hypothesis Test with a standard deviation or variance k - 1 Goodness-of-Fit with k categories (r - 1)(c - 1) Contingency Table with r rows and c columns k - 1 Kruskal-Wallis test with k samples 8056_Barrelfold_pp01-08.indd 4 9/26/12 9:52 AM