Lecture 13-4720 PDF

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Fundamentals of Financial Calculations (Engineering Econ.) Samantha L. Fall 2021' Chapter # 13...

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TSYH 4720 Fundamentals of Financial Calculations

Lecture 13: Distribution of the Sample Means You want to estimate the mean age of 1st-year BCIT students. The distribution of ages is skewed right. You randomly select 100 students and ask them how old they are. From your sample you obtain a mean age of 21.3 years. You select another 100 students and get a mean of 22.4. → There are many different samples of size 100 that can be taken from this population. 1. The distribution of the ages of first-year students at BCIT is highly skewed to the right with a mean age of 22 years and a standard deviation of 2.8 years. (a) If you randomly select a student find the probability he/she is from 21 to 23 years old. (b) You randomly ask 49 1st year students how old they are. Find the mean and the standard deviation of the sample means. Describe the shape of the sampling distribution. (c) What is the probability of obtaining a sample mean age between 21 and 23 years? 2. You recently graduated from the BCIT Financial Management program. You just started working as an accountant in a large CA firm. Your boss claims that the average starting salary for an accountant who just graduated is $30,000/year so he offers to pay you $30,000. The standard deviation is $3,000/year and salaries are approximately normally distributed. (a) What percentage of accountants earn at least $32,000/year just after graduating? (b) If you randomly select an accountant, what is the probability he earns at least $32,000? (c) You randomly select 40 accountants who recently graduated and ask them how much they earn. What is the probability of obtaining an average salary of at least $32,000? 3. The weight of adults in a Seattle is normally distributed with a mean of 170 pounds and a standard deviation of 24 pounds. On a raft that takes people across the river, a sign states, “Maximum capacity 3,200 pounds or 16 adults". What is the probability that a random sample of 16 adults will exceed the weight limit of 3,200 pounds? 1. (a) n = 1, the population is not normally distributed→ cannot answer 2. (a) 25.14% (b) 0.2514

(c) almost 0

(c) 0.9876

3. almost 0

Lab Exercises: Textbook Reading 7.1, 7.3, 7.4 1. The amount of time per day spent by adults watching TV is normally distributed with a mean of 5 hours and a standard deviation of 1.5 hours. (a) What is the probability a randomly selected adult watches TV for more than 6 hours/day? You randomly select 9 adults and record how long they spend per day watching TV: (b) What is the probability that the mean time of the 9 adults spent watching TV is more than 6 hours? (c) Find the probability the total time spent watching TV for the 9 adults exceeds 36 hours. (d) What is the minimum sample mean TV watching time of the 9 adults for the top 3% ?

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TSYH 4720 Fundamentals of Financial Calculations

2. The average monthly rent for students in a college town is normally distributed with a mean of $650 and a standard deviation of $80. (a) What is the probability that the total revenue from renting 10 randomly selected apartments falls between $6,000 and $7,000? (b) A random sample of 100 students who rent apartments was taken. What is the probability of getting a sample mean monthly rent that exceeds $660? (c) Let’s assume the population mean monthly rent was unknown, but the standard deviation is known to be $80. A sample of 100 rentals was selected to estimate the mean monthly rent paid by the whole student population. What is the probability that the sample mean rent differs from the actual mean rent by more than $10? How about more than $20? 3. WestJet is concerned about the amount of baggage its passengers are bringing on the flight. A study last year of passenger bags revealed that the bags had a mean weight of 18.2 kg and a standard deviation of 3.75 kg and that the weights are approximately normally distributed. (a) 1 out of every 40 bags should weigh more than what weight? (b) The maximum bag weight limit is 23 kg. What percentage of bags exceed this weight? (c) The baggage supervisor has his staff randomly select 9 bags and weigh them. (i) What is the probability the sample mean weight will be less than 15 kg? (ii) What is the probability the sample mean will be within 2 kg of the population mean? (d) A small plane with 80 seats can handle 1,600 kg of checked baggage. What is the probability that 80 passengers will check too much baggage for the plane to handle? 4. The mean weight of Canadian men is 180 pounds with a standard deviation of 30 pounds. The weights are approximately normally distributed. The elevator in a men’s athletic club has a capacity of 10 people and a maximum load of 2,400 pounds. What is the probability that if 10 men get on the elevator, they will overload its weight limit? 5. There are 36 students in your instructor’s night school statistics class. From experience she knows that the time needed to mark an exam averages 5 minutes with a standard deviation of 3 minutes. Your instructor starts marking at 4:15 pm and hopes to finish marking the 36 exams before the Amazing Race show starts at 8:00 pm. What is the probability that she does not finish marking before 8:00 pm and has to miss part of the show? Hint: First calculate how fast, on average, she needs to mark an exam if she wants to finish by 8:00 pm. 6. The mean grade point average (GPA) for students at a university is 3.0 and the standard deviation is 0.375. The GPAs are approximately normally distributed. 9 students are randomly sampled. The top 10% of sample mean GPA’s will be higher than what GPA? 7. A sign on a vending machine that dispenses coffee states that each cup contains 200 ml of coffee. The machine actually dispenses a mean amount of 208 ml per cup and the standard deviation is 9 ml. The amount of coffee dispensed is normally distributed. If the machine is used 300 times, how many cups would you expect to contain less than the amount stated?

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TSYH 4720 Fundamentals of Financial Calculations

8. The time taken by students to finish a statistics final exam is normally distributed with a mean of 96 minutes with a standard deviation of 20 minutes. Students are given two hours to write the exam and they are not permitted to leave during the last 10 minutes. If 500 students write the exam, how many students would you expect to leave the exam before the end? Assume all students who finish before the last 10 minutes leave the exam room. 9. Students must write a Scholastic Assessment Test (SAT) when applying to a US college. SAT scores are normally distributed with a mean of 500 and a standard deviation of 100. (a) A college decides to only admit students with SAT scores of 550 or higher. If the college admitted 617 students, how many students were rejected? (b) Suppose you select 16 students at random. What is the probability of the average SAT score of the 16 students being 510 or less? 10. Brewers Association studies shows that the average Canadian adult drinks 90 litres of beer per year. The number of litres of beer drank is highly skewed to the right with a standard deviation of 48 litres. A random sample of 36 Canadian adults of legal drinking age is taken. (a) What is the mean and standard deviation of the sample means? Describe the shape. (b) What is the probability of our sample having a mean greater than 100 litres? (c) What is the probability of our sample having a mean less than 75 litres? (d) What is the probability of our sample having a mean between 75 and 100 litres? (e) Find the probability of our sample mean being within ten litres of the population mean. (f) Find the minimum sample mean consumption needed to be in the top 1% of all samples. 11. Jean-Betty Crumpler wants to estimate the average annual household income for adults living in the lower mainland. From past data the standard deviation is thought to be $12,000. (a) She randomly selects 100 households. Find the probability that her estimate of the mean income (sample mean) is within one $1,000 of the actual mean (pop mean) income? (b) Find the probability that the sample mean income for a random sample of 576 households is within $1,000 of the population mean income (i.e. the sampling error is at most $1,000). Would you recommend a sample size of 100 or 576 if you want to have at least a 95% chance that the sample mean is within $1,000 of the population mean? 12. An elevator company has conducted a study and determined the average adult weight is 65 kg with a standard deviation of 15 kg. The distribution is approximately normally distributed. (a) Find the probability of an individual weighing less than 44 kg. (b) Find the probability of an individual weighing more than 95 kg. (c) The company has decided that the elevator will have a capacity of 5 people and a maximum load of 475 kg. What is the probability that load will be exceeded if there are 5 people in the elevator? (d) The company also makes another elevator that can hold 10 people and has a maximum capacity of 950 kg. Find the probability that 10 people will exceed the maximum load. Page 3 of 4

TSYH 4720 Fundamentals of Financial Calculations

13. How much is the standard deviation of the sample means (standard error of the mean) reduced if the sample size is 4 times bigger? 100 times bigger? 14. KPL, a CPA firm, performs audits for the Precision Tool Co. Precision tells the auditor in charge that the average balance of the 20,000 accounts receivables is $15,000 with a standard deviation of $5,000. As part of the audit, the auditor selected a random sample of 400 accounts from the 20,000 accounts receivable on Precision’s books. (a) Find the mean and standard deviation of the sample means and describe the shape. (b) Suppose the auditor’s sample of 400 account receivables produced a mean of $14,925. Do you believe Precision’ claim that the mean is $15,000? Hint: Find the probability that the sample mean would be $14,925 or less when the actual mean is $15,000. (c) Suppose the sample mean was $14,000 instead. Now do you believe Precision’ claim? Solutions: 1. (a) 0.2514 (b) 0.0228 (c) 0.9772 (d) 2. (a) 0.9522 (b) 0.1056

x = 5.94 hours

(c) 2×0.1056 = 0.2112, 2×0.0062 = 0.0124

3. (a) 25.55 kg (b) 10% (c) (i) 0.0052 (ii) 0.8904 (d) Z = 4.29→ ~ 0% 4. Z = 6.32 → ~ 0 5. 0.0062 6. 3.16 9. (a) 1,383 = 2,000 – 617 (b) 0.6554

7. 56 cups

8. 379

10. (a) µ x = µ = 90 litres σ x = 8.0 litres , the distribution of the sample means is ~ normal (b) 0.1056 (c) 0.0301 (d) 0.8643 (e) 0.7888 (f) 108.64 litres 11. (a) 0.5934 (b) 0.9544 There is a 95.44% chance of obtaining a sample mean that is within $1,000 of the pop mean (> 95% chance the sampling error is < $1,000) with a sample size of 576 12. (a) 0.0808

(b) 0.0228

(c) z = 4.47 so almost 0

(d) z = 6.32 so almost 0

13. For a sample size 4 times bigger, the standard error is ½ as big. 100 times bigger, 1/10 as big. 14. (a) µ x = µ = $15,000 σ x = $250 , the distribution of the sample means is ~ normal (b) It is believable. There is a 38.21% chance of getting a sample mean of $14,925 or less (c) Almost no chance of getting a sample mean of $14,000 or less if the pop mean is $15,000. Precision’s claim is not believable. ($14,000 is 4 standard deviations below the pop mean)

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