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ECON1005 Introductory Statistics Lecture Exercises Workbook
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The University of the West Indies Department of Economics ECON1005 – Introductory Statistics Lecture Exercises Workbook Prepared by: Dr K. Dixon Hamil Version Date: August 2023 Table of Contents Week 2 … … … … … … … … … … 1 Week 3 … … … … … … … … … … 9 Week 4 … … … … … … … … … … 16 Week 5 … … … … … … … … … … 21 Week 6 … … … … … … … … … … 26 Week 7 … … … … … … … … … … 30 Week 8 … … … … … … … … … … 33 Week 9 … … … … … … … … … … 39 Week 10 … … … … … … … … … … 45 Week 11 … … … … … … … … … … 49 … … … … … … … 55 Normal Distribution Table ECON1005 Lecture Exercises – Week 2 Lecture Exercises Questions – Unit 1 & Units 2.1 – 2.2 These extra questions will be worked in class during Week 2 of the semester. They are based on the concepts in the videos related to Unit 1 and Unit 2.1 (Introduction to Data Collection, Graphical Methods for Representing Data & Frequency Distributions). USE THE FOLLOWING SCENARIO TO ANSWER QUESTIONS 1 - 5 Researchers are interested in determining the amount of money first year UWI Students spend on textbooks in Semester I. They randomly selected 500 students from three randomly selected faculties – Medicine, Social Science and Science & Technology. The average cost was determined to be $30,000. 1. The information collected from each person in the sample is A. The average money spent on all textbooks for Semester I B. The total money spent on textbooks for Semester I C. The average money spent on all textbooks for the academic year D. The total money spent on all textbooks for the academic year 2. $30,000 is a/an A. Parameter B. Statistic C. Ordinal variable D. Discrete variable E. Interval variable 3. Faculty is a/an A. Continuous variable B. Ordinal variable C. Nominal variable D. Interval variable E. Ratio variable 1 ECON1005 Lecture Exercises – Week 2 4. The best graphical method to represent the number of students selected from each faculty is a A. Histogram B. Stem-and-leaf plot C. Pie Chart D. Less than cumulative frequency ogive E. More than cumulative frequency ogive 5. The best graphical method to represent the amount of money in this case is A. Histogram B. Stem-and-leaf plot C. Pie Chart D. Less than cumulative frequency ogive E. More than cumulative frequency ogive 2 ECON1005 Lecture Exercises – Week 2 USE THE FOLLOWING TO ANSWER QUESTIONS 6 - 9 A study was conducted to determine the average GPA that a student graduating from The UWI in 2020 obtained. Two hundred of the students from the graduating class were selected for more in-depth study Match each of the following with the SINGLE choice that best describes the given situation. Each answer choice may be used once, more than once or not at all. A. Census B. Parameter C. Population E. Statistic F. Inferential Statistics G. Descriptive Statistics D. Sample 6. The average GPA for all graduating students in 2020. 7. A histogram of the GPAs for the sample graduating students. 8. The average age of the 200 students. 9. The average GPA of the 200 students was 2.50. Using this, the researchers said that the average GPA of all students from the 2020 graduating class was 2.50. 3 ECON1005 Lecture Exercises – Week 2 USE THE FOLLOWING TO ANSWER QUESTIONS 10 - 12 A. Simple Random Sample B. Stratified Random Sample D. Multistage Sample E. Voluntary Sample C. Cluster Sample 10. An international football club manager is interested in selecting Jamaican players for his team. Not wanting to show bias he randomly selects 3 players from each of the professional football teams in each of the 14 parishes (Jamaica has 14 parishes). 11. An international football club manager is interested in selecting players from Trinidad and Tobago for his team. Not wanting to show bias he randomly selects 3 players from each of 4 randomly selected regions in the country (Trinidad and Tobago has 9 regions). 12. An international football club manager is interested in selecting players from Barbados for his team. Not wanting to show bias he randomly selects 3 parishes and uses every player on the teams in those parishes (Barbados has 11 parishes). Simple Random Sample Stratified Sample 4 ECON1005 Lecture Exercises – Week 2 Cluster Sample Source: https://faculty.elgin.edu/dkernler/statistics/ch01/1-3.html Mutlistage Sample Adapted from: https://faculty.elgin.edu/dkernler/statistics/ch01/1-3.html 5 ECON1005 Lecture Exercises – Week 2 QUESTION 13 – 14 RELATE TO THE DIAGRAM BELOW The following is a graph of the scores on an aptitude test. 13. If the pass mark is 50%, how many students passed the test? A. 15 B. 20 C. 35 D. 50 14. How many students scored between 50% and 70% on this test? A. 15 B. 20 C. 35 D. 50 6 ECON1005 Lecture Exercises – Week 2 USE THE FOLLOWING SCENARIO TO ANSWER QUESTIONS 15 - 16 The following represents the cumulative relative frequency graph for the heights of 50 patients. 15. The class limit that has the largest frequency is A. 69.5 – 88.5 B. 70 – 88 C. 107.5 – 126.5 D. 108 – 126 16. The cumulative frequency for the fourth class is A. 5 B. 15 C. 35 D. 40 7 ECON1005 Lecture Exercises – Week 2 17. Complete the table below which shows the frequencies of the income for 30 employees at a local small business (in $1000s). Income Number of Employees 26 to less than 28 2 28 to less than 30 11 30 to less than 32 8 32 to less than 34 5 34 to less than 36 4 Midpoints 8 Upper Boundaries More than Cumulative Relative Freq. ECON1005 Lecture Exercises – Week 3 Lecture Exercises – Unit 2.2 & Unit 2.3 These lecture exercises will be worked in class during Week 3 of the semester. They are based on the concepts in the videos related to Unit 2.2 and Unit 2.3 (Measures of Shape & Numerical Methods of Describing Data). 1. The back to back stem and leaf plot below shows the exam grades (out of 100) of two different classes. The best way to describe this data is that A. Class Period 1 is left skewed and Class Period 2 is right skewed. Both are unimodal B. Class Period 1 is right skewed and Class Period 2 is left skewed. Both are unimodal C. Both Class Period 1 and Class Period 2 are symmetric and unimodal D. Class Period 1 is left skewed and Class Period 2 is right skewed. Both are bimodal E. Class Period 1 is right skewed and Class Period 2 is left skewed. Both are bimodal F. Both Class Period 1 and Class Period 2 are symmetric and bimodal 9 ECON1005 Lecture Exercises – Week 3 2. A sample of first year students were asked to record how many hours per week they spent studying ECON1005. The results are presented in the table below. Hours Spent Studying No. of Students 2 6 3 6 4 4 5 9 6 3 7 2 a. What is the average number of hours per week spent studying? 10 ECON1005 Lecture Exercises – Week 3 b. What is the median number of hours per week spent studying? Gonick, Larry, and Woollcott Smith. The cartoon guide to statistics. HarperCollins Publishers, Inc, 1993. 11 ECON1005 Lecture Exercises – Week 3 c. What is the standard deviation of hours per week spent studying? Gonick, Larry, and Woollcott Smith. The cartoon guide to statistics. HarperCollins Publishers, Inc, 1993. 12 ECON1005 Lecture Exercises – Week 3 d. How would you describe the shape of this dataset? 13 ECON1005 Lecture Exercises – Week 3 3. A substitute teacher was asked to keep track of how long it took her to get to her assigned school each morning. Here is a stem plot of the data. For this dataset you would expect: A. The mean and the median to be approximately equal B. The mean to be greater than the median C. The mean to be less than the median D. We cannot determine the relationship between the mean and the median 4. Using the graph below, what do A, B and C represent? A. Mode, Median, Mean B. Mean, Median, Mode C. Median, Mode, Mean D. Mean, Mode, Median 14 ECON1005 Lecture Exercises – Week 3 5. The following histogram represents the number of days of school missed due to poor weather conditions. The best measures to use to describe this data are: A. Median and Interquartile Range B. Mean and Standard Deviation C. Median and Standard Deviation D. Mean and Interquartile Range Gonick, Larry, and Woollcott Smith. The cartoon guide to statistics. HarperCollins Publishers, Inc, 1993. 15 ECON1005 Lecture Exercises – Week 4 Lecture Exercises – Unit 2.3 & Units 3.1-3.2 These lecture exercises will be worked in class during Week 4 of the semester. They are based on the concepts in the videos related to Unit 2.3 (Numerical Methods of Describing Data) and Units 3.1-3.2 (Probability). 1. The weights of males and females are shown in the boxplots below. Select the answer which is correct. A. 25% of the female students weigh more than approximately 130 pounds B. The weight of male students is less variable than the weight of female students C. Female students have the higher median Gonick, Larry, and Woollcott Smith. The cartoon guide to statistics. HarperCollins Publishers, Inc, 1993. 16 ECON1005 Lecture Exercises – Week 4 2. You have an equally likely chance of choosing any number from 1 to 10. What is the probability that you choose a number greater than 6? A. 1/10 B. 2/5 C. 1/2 D. 3/5 17 ECON1005 Lecture Exercises – Week 4 3. If A and B are two events such that P(Ac) = 0.4 and P(A ∩ B) = 0.2, then P(A ∩ Bc) is equal to A. 0.2 B. 0.4 C. 0.6 D. 0.8 https://collegemathteaching.files.wordpress.com/2011/05/dontknowmath.gif 18 ECON1005 Lecture Exercises – Week 4 4. Three persons are randomly selected each year to serve as President, Vice President and Treasurer of a club. Suppose there are 15 males and 10 females in this club, all who are eligible to be selected. What is the probability that 2 males and 1 female are selected to serve for this coming year? A. 0.0200 B. 0.1200 C. 0.2137 D. 0.4565 19 ECON1005 Lecture Exercises – Week 4 5. You and your friend Davia just graduated with your undergraduate degrees in Statistics. Both of you apply for jobs at two unrelated companies. The probability that you are selected for an interview is 2/5 while the probability of Davia being selected for an interview is 4/7. What is the probability that both of you are selected to interview? A. 8/35 B. 34/35 C. 27/35 D. 6/35 6. A sales manager at a car dealership has determined that selling a Porsche Cayenne and a Range Rover Sport on the same day is not possible. The probability of selling a Porsche Cayenne is 0.6, while the probability of selling a Range Rover Sport 0.01. He will get a $100,000 bonus if he sells any one of these two cars on a given day. What is the probability that he gets the bonus next week Thursday? A. 0.006 B. 0.590 C. 0.610 D. 0.700 20 ECON1005 Lecture Exercises – Week 5 Lecture Exercises – Units 3.3 & 3.4 These lecture exercises will be worked in class during Week 5 of the semester. They are based on the concepts in the videos related to Units 3.3 and 3.4 (Probability). 1. Five hundred employees were selected from a city’s large private companies and they were asked whether or not they have any retirement benefits provided by their companies. Based on this information, the following table was prepared. An employee is selected at random from these 500 employees. a) What is the probability that the employee is a woman? b) What is the probability that the employee has retirement benefits? c) If a man is selected, what is the probability that he has retirement benefits? 21 ECON1005 Lecture Exercises – Week 5 d) If a woman is selected, what is the probability that she does not have retirement benefits? e) What is the probability that the selected employee is female and has retirement benefits? f) Are the events “man” and “yes” mutually exclusive? Why or why not? g) Are the events “woman” and “yes” independent? Why or why not? 22 ECON1005 Lecture Exercises – Week 5 2. The probability function of a random variable X is defined as: X -1 -2 0 1 2 f(x) k 2k 3k 4k 5k a) Find the value of k b) What is the probability that X is larger than 0. 23 ECON1005 Lecture Exercises – Week 5 c) Calculate the expected value of this distribution. d) Calculate the standard deviation of this distribution. 24 ECON1005 Lecture Exercises – Week 5 3. A charity fundraiser has a Spin the Pointer game that uses as spinner like the one below. A donation of $2 is required to play the game. For each $2 donation, a player spins the pointer once and receives the amount of money indicated in the sector where the pointer lands on the wheel. The spinner has an equal probability of landing in each of the 10 sectors. a) If X represents the net contribution of the charity when one person plays the game once, complete the table for the probability distribution of X. X $2 $1 -$8 p(x) b) What is the expected value of the net contribution to the charity for one play of the game? 25 ECON1005 Lecture Exercises – Week 6 Lecture Exercises – Units 3.5 & 3.6.1 These lecture exercises will be worked in class during Week 6 of the semester. They are based on the concepts in the videos related to Units 3.5 (Binomial Distribution) and 3.6.1 (Normal Distribution Part 1). 1. Shuffle a deck of 52 cards. Turn over the top card and note its value. Put the card back in the deck and shuffle again. You repeat this procedure 15 times. Let X be the random variable representing the number of face cards you observe. Is this X a binomial random variable? 26 ECON1005 Lecture Exercises – Week 6 2. A coin is tossed four times. Calculate the probability of obtaining more heads than tails. 3. There is an 80% chance that a patient with a certain disease will be successfully treated with a new medical treatment. Suppose that the treatment is used on 40 patients. What is the average number of patients who will be successfully treated? A. 8 B. 20 C. 32 D. 40 27 ECON1005 Lecture Exercises – Week 6 https://i.ytimg.com/vi/Txylq6RLiK8/maxresdefault.jpg 4. A 12oz can of soda has a mean volume of 12oz, with a standard deviation of 0.25oz. Assuming that the volume of soda in a can is normally distributed, determine the probability of a can having less than 11.5oz of soda. A. 0.0015 B. 0.0250 C. 0.0225 D. 0.0235 28 ECON1005 Lecture Exercises – Week 6 5. A machine produces electrical components. 99.7% of the components have lengths between 1.176cm and 1.224cm. Assuming that this data are normally distributed, what are the mean and standard deviation? A. Mean = 1.210 and S.D. = 0.008 B. Mean = 1.200 and S.D. = 0.004 C. Mean = 1.190 and S.D. = 0.008 D. Mean = 1.200 and S.D. = 0.008 E. Mean = 1.210 and S.D. = 0.004 6. The GPA of first year college students is normally distributed with a mean of 2.9 and standard deviation of 0.6. What is the GPA of the highest 2.5% of the students? 29 ECON1005 Lecture Exercises – Week 7 Lecture Exercises – Units 3.6.2 & 3.6.3 These lecture exercises will be worked in class during Week 7 of the semester. They are based on the concepts in the videos related to Units 3.6.2 and 3.6.3 (Normal Distribution Part 2). https://mathwithbaddrawings.com/2018/08/15/the-bubble-under-the-mathematical-rug/ 1. Private colleges tend to cost more than public colleges. Here are some summary statistics for the amount of college debt that undergraduate students graduate with from institutions in one region. Institution Type Private Mean Debt ($’000) $24 Standard Deviation ($’000) $16 Public $18 $15 Nicholas graduated from a private institution and Tarique graduated from a public institution. Each had $35,000 of college debt. Relative to their institution type, who graduated with more debt? A. Nicholas B. Tarique C. Nicholas and Tarique graduated with equally as much debt relative to their institution types D. It is impossible to say without seeing all of the individual debt amounts E. It is impossible to say since we do not know the shape of either distribution. 30 ECON1005 Lecture Exercises – Week 7 2. Let X represent the distances travelled to work by employees at a large company. Assuming that X ~ N(30, 8), find the probability that a randomly selected employee has to travel more than 20km to work. 31 ECON1005 Lecture Exercises – Week 7 3. The mean inside-diameter of washers produced by a machine is 0.502 inches and the standard deviation is 0.005 inches. The purpose for which these washers are intended allows a maximum tolerance in the inside-diameter of 0.496 to 0.508 inches, otherwise the washers are considered defective. Determine the percentage of defective washers produced by the machine, assuming the inside-diameters are normally distributed. 32 ECON1005 Lecture Exercises – Week 8 Lecture Exercises – Units 3.6.4 & 3.7 These lecture exercises will be worked in class during Week 8 of the semester. They are based on the concepts in the videos related to Unit 3.6.4 & 3.7 (Normal Distribution – Backwards Normal & the Normal Approximation to the Binomial Distribution). 1. A machine produces chocolate bars whose weight is normally distributed with mean 200g and standard deviation 8g. The company will not allow more than 10 % of bars to be over a certain weight. Where should the company set the limit? 33 ECON1005 Lecture Exercises – Week 8 2. The distribution of annual profit at a chain of stores was approximately normal with mean $66,000 and standard deviation $21,000. The executives conducted an audit of the stores with the lowest 20% of profits. What is the closest to the maximum annual profit at a store where the executives conducted an audit? A. $40,000 B. $45,000 C. $48,000 D. $56,000 E. $84,000 34 ECON1005 Lecture Exercises – Week 8 3. The distribution of average wait times in drive-through restaurant lines in one town was approximately normal with mean 185 seconds and standard deviation 11 seconds. A journalist wrote an article about the restaurants whose average drive-through wait times were in the top 25% for that town. What is the minimum average wait time for restaurants that the journalist included in the article? A. 178 seconds B. 193 seconds C. 194 seconds D. 196 seconds E. 202 seconds 35 ECON1005 Lecture Exercises – Week 8 4. The distribution of annual profit at a chain of stores was approximately normal with mean $66,000 and standard deviation $22,000. The stores with profits in the top 5% each had a reward party for the employees to celebrate. What is closest to the minimum annual profit for a store that had a reward party? A. $30,000 B. $60,000 C. $98,000 D. $102,000 E. $109,000 36 ECON1005 Lecture Exercises – Week 8 5. You select a random sample of 1600 tyres from an ongoing production process in which 8% of all such tyres produced are defective. a) What is the probability that 150 or fewer tyres will be defective? b) Approximate the probability of getting exactly 150 defective tyres. Gonick, Larry, and Woollcott Smith. The cartoon guide to statistics. HarperCollins Publishers, Inc, 1993. 37 ECON1005 Lecture Exercises – Week 8 38 ECON1005 Lecture Exercises – Week 9 Lecture Exercises – Units 4.1 – 4.3.1 These lecture exercises will be worked in class during Week 9 of the semester. They are based on the concepts in the videos related to Units 4.1 and 4.2 (Sampling Distributions, Confidence Intervals for the Mean). 1. The monthly cost for telephone service for all households in a large city has a skewed distribution with a mean of $96 and a standard deviation of $27. If 𝑥̅ is the average monthly cost for telephone service in a sample of 90 households in this city, find the mean and standard deviation of the sampling distribution of the mean. Also describe the shape of this distribution. 39 ECON1005 Lecture Exercises – Week 9 2. Let the random variable X follow a normal distribution with a mean µ and standard deviation σ. Let 𝑥̅1 and 𝑥̅2 be the means of two randomly and independently selected samples of sizes with n1 = 16 and n2 = 25, respectively. Given the following two expressions, which of the statements below is true? (1) 𝑃(𝑋̅1 < 𝜇) (2) 𝑃(𝑋̅2 < 𝜇) A. The value of (1) is greater than the value of (2). B. The value of (2) is greater than the value of (1). C. The value of (1) is equal to the value (2). D. Unable to determine the relationship between (1) and (2). 3. The balances of all savings accounts at a local bank have a distribution that is skewed right with mean $12,450 and standard deviation $4,300. Find the probability that the average balance of a sample of 50 savings accounts selected from this bank will be a) more than $11,500 40 ECON1005 Lecture Exercises – Week 9 41 ECON1005 Lecture Exercises – Week 9 b) between $12,000 and $13,800 42 ECON1005 Lecture Exercises – Week 9 4. A publishing company has just published a new college textbook. Before the company decides the price at which to sell this textbook, it wants to know the average price of all such textbooks in the market. The research department at the company took a sample of 25 comparable textbooks and collected information on their prices. This information produced a mean price of $145 for this sample. It is known that the standard deviation of the prices of all such textbooks is $35 and the population of such prices is normal. The aim is to construct a 90% confidence interval for the price of these types of textbooks. a) What is the value at the centre of this confidence interval? b) What is the critical value for this confidence interval? c) What is the standard error of this confidence interval? d) What is the margin of error of this interval? 43 ECON1005 Lecture Exercises – Week 9 e) What are the lower and upper limits of this interval? f) Does this interval apply to the sample or the population? 44 ECON1005 Lecture Exercises – Week 10 Lecture Exercises – Unit 4.3 These lecture exercises will be worked in class during Week 10 of the semester. They are based on the concepts in the videos related to Unit 4.3 (Inferences for Population Means). 1. Researchers claim that 60 is the average number of tissues a person uses during the course of a cold. The company who makes Kleenex brand tissues thinks that fewer of their tissues are needed. What are their null and alternative hypotheses? A. 𝐻0: 𝜇 = 60 𝑣𝑠 𝐻𝑎: 𝜇 > 60 B. 𝐻0: 𝜇 = 60 𝑣𝑠 𝐻𝑎: 𝜇 < 60 C. 𝐻0: 𝑥̅̅ = 60 𝑣𝑠 𝐻𝑎: 𝑥̅̅ < 60 D. 𝐻0: 𝜇 < 60 𝑣𝑠 𝐻𝑎: 𝜇 = 60 2. The statement “If there is sufficient evidence to reject a null hypothesis at the 10% significance level, then there is sufficient evidence to reject it at the 5% significance level” is: Please select the best answer of those provided below. A. Always True B. Never True C. Sometimes True; the p-value for the statistical test needs to be provided for a conclusion D. Not Enough Information; this would depend on the type of statistical test used 45 ECON1005 Lecture Exercises – Week 10 3. The owner of a local nightclub recently surveyed a random sample of 300 customers of the club. She would like to determine whether or not the mean age of her customers is over 35. If so, she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will be made. Suppose she found that the sample mean was 35.5 years and population standard deviation was 5 years. a) What is the p-value associated with the test statistic? A. B. C. D. 0.0418 0.9572 0.0421 0.0836 46 ECON1005 Lecture Exercises – Week 10 b) Based on your results, should she make any changes to the entertainment? A. B. C. D. Yes, because there is evidence that the population average age of her customers is over 35. Yes, because there is evidence that the sample average age of her customers is over 35. No, because there is no evidence that the population average age of her customers is over 35. No, because there is no evidence that the sample average age of her customers is over 35. 47 ECON1005 Lecture Exercises – Week 10 4. Your investment advisor proposes that you invest in a monthly income investment plan that promises a variable return each month. You will invest in it only if you are assured of a return no less than an average of $180 per month. Your advisor also tells you that for the past 300 months, the scheme had an average investment return of $190. It is known that all investments in such a scheme has a standard deviation of $75. Should you invest in this scheme? Why? 48 ECON1005 Lecture Exercises – Week 11 Lecture Exercises – Unit 4.4 These lecture exercises will be worked in class during Week 11 of the semester. They are based on the concepts in the videos related to Unit 4.4 (Inferences for Population Proportions). 1. A simple random sample of 500 parents were asked if they were in favour of resuming face-to-face classes for their children. If it is known that in fact 45% of parents are in favour of this action, what is the probability that more than half of the sample are in favour of this action? 49 ECON1005 Lecture Exercises – Week 11 2. Recently, it was reported that 71.9% (363) of 505 randomly selected university students that were in their first year of university in 2010 returned in 2011 for their second year. a) Estimate the national retention rate for university students using a 95% confidence interval. 50 ECON1005 Lecture Exercises – Week 11 b) Choose the correct answer to the following statement: If we calculated a 98% confidence interval instead of a 95% confidence interval, the 98% confidence interval would be A. narrower B. wider c) Choose the correct interpretation of the 95% confidence interval in part a): A. In 95% of all random samples of universities, the retention rate will be 71.9%. B. In 95% of all random samples of universities, the retention rate is between 68.0% and 75.8%. C. There is a 95% chance that the true retention rate of universities is in the interval (68.0%, 75.8%) D. If many random samples are selected, each sample with 505 university students, 95% of the sample retention rates 𝑝̂ will be in the interval (0.680, 0.758). 51 ECON1005 Lecture Exercises – Week 11 3. A national health organization warns that 30% of the high school students nationwide have been drunk. Concerned, a local health agency wants to determine the percentage of high school students in their city that have been drunk. Suppose the local health agency does not want to take a guess as to what the true percentage maybe, how many high school students do they need to survey to achieve a margin of error of 5%? 52 ECON1005 Lecture Exercises – Week 11 4. Suppose we think the casino is cheating by using dice that do not sum to seven as often as they should. That is, if we have two dice (each six-sided with numbers 1 to 6), we expect that a sum of seven should occur 1/6 of the times. We collect data on 1000 dice rolls and find that 153 of them sum to seven. Is this enough evidence to accuse the casino of cheating? A. Yes, because there is evidence that the population proportion of die summing to 7 is not equal to 1/6. B. Yes, because there is evidence that the sample proportion of die summing to 7 is not equal to 1/6. C. No, because there is no evidence that the population proportion of die summing to 7 is not equal to 1/6. D. No, because there is no evidence that the sample proportion of die summing to 7 is not equal to 1/6. 53 ECON1005 Lecture Exercises – Week 11 54
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