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Variability and Statistical Analysis SENIOR 4 APPLIED MATHEMATICS EXERCISES Exercise 1: Mean and Standard Deviation 1. For each of the following sets of data calculate the mean, median, mode and standard deviation. A 45 45 47 47 55 63 63 65 65 B 15 26 40 49 55 55 75 82 98 C 4 11 27 33 52 60 83 85 140 2. For each of the sets of information given in the following tables use the calculator to find the mean, median, mode and standard deviation. Late Prehistoric Arrowhead Measurements (in mm) Big Goose Creek (52 arrowheads) Length Width 16 16 17 17 18 18 18 18 19 20 20 21 21 21 22 22 22 23 23 23 24 24 25 25 25 26 26 26 26 27 27 27 27 27 28 28 28 28 29 30 30 30 30 30 30 31 33 33 34 35 39 40 10 11 11 11 11 11 12 12 12 12 12 12 12 12 13 IS 13 13 13 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 14 14 14 14 14 14 15 15 15 15 15 15 16 17 18 5 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 11 11 11 12 12 13 Neck Width DAKOTA COLLEGIATE Page 1 of 10 Variability and Statistical Analysis SENIOR 4 APPLIED MATHEMATICS EXERCISES 3. A class of 30 students received the following marks in a mathematical examination. Use technology to find the mean, median, mode and standard deviation of this set of scores. 78 92 62 52 65 59 53 63 68 73 71 63 69 74 73 81 55 71 75 81 84 77 80 75 41 57 91 62 65 49 4. Potatoes are sold at E and S Supermarket in 5 kg bags. Ten bags are selected at random to provide an estimate of the mean and standard deviation of all the bags sold at this supermarket. The 10 bags weighed the following (in kg): 5.4, 5.4, 5.3, 5.2, 5.3, 5.3, 5.1, 5.0, 4.9, and 5.1. Give the estimates for the mean and standard deviation obtained. 5. Calculate the mean and the standard deviation for the following data, based on the heights of 100 Senior Years students. height 153.5 to 160.5 160.5 to 167,5 167.5 to 174.5 174,5 to 181.5 181.5 to 188.5 TOTALS class mark 157 164 171 178 185 frequency 5 16 43 27 9 100 6. Calculate the mean and the standard deviation for the following data, based on the weights of 125 newborn infants, in pounds. weight 3.5 to 4.5 4.5 to 5.5 5.5 to 6.5 6.5 to 7.5 7.5 to 8.5 8.5 to 9.5 9.5 to 10.5 10.5 to 11.5 TOTALS DAKOTA COLLEGIATE class mark 4 5 6 7 8 9 10 11 frequency 4 11 19 33 29 17 8 4 125 Page 2 of 10 SENIOR 4 APPLIED MATHEMATICS EXERCISES Variability and Statistical Analysis 7. Three manufactures of toasters claim that the average life of their toasters, under normal use, is five years. A consumer’s group decided to test each company’s claim. It compiles the following list, in years, of toasters manufactured by each: Company X: 0.5, 1.6, 2, 3.5, 4, 4.5, 6, 7, 7.9, 8, 10 Company Y: 4, 4, 5, 5, 5, 6, 11, 13, 14, 15, 16 Company Z: 2, 3, 4, 4, 6, 13, 14, 15 a. Which “average” was each company using to support its claim? b. From which company would you buy a toaster? Why? DAKOTA COLLEGIATE Page 3 of 10 SENIOR 4 APPLIED MATHEMATICS EXERCISES Exercise 2: 1. Variability and Statistical Analysis The Normal Distribution and the Approximation to the Normal Distribution. Refer to question 1, Ex. 1. a. Give the boundaries of one standard deviation above and below the mean. ( , ) b. Find the percent of data lying within one standard deviation of the mean. ( , ) c. Is this distribution considered normal? 2. Refer to question #2, Ex. 1. Find the percent of scores lying within one standard deviation of the mean. ( , ). Do this for each of the sets. Are the distributions considered “normal”? 3. A machine packaging candy in 90 gram packages is thought to be faulty. Ten packages were randomly selected, and the actual masses, in grams are: 86, 91, 89, 92, 90, 93, 90, 90, 91, 88 If the spread of the masses is too great, the machine is considered to be faulty. Statistically, we use the standard deviation to judge this. In this case, if the standard deviation of the set of scores is greater than 1.3, the machine is considered faulty and will need adjustment or repair. Is the machine faulty? How did you determine the answer? 4. Using a graphing calculator, find the mean and standard deviation of the following frequency distribution. Find the percentage of scores that lie within 1 standard deviation of the mean. Explain why this is or is not a normal distribution. Scores 0 1 2 3 4 5 6 7 8 9 DAKOTA COLLEGIATE Frequency 80 60 80 70 60 80 50 60 70 60 Page 4 of 10 Variability and Statistical Analysis SENIOR 4 APPLIED MATHEMATICS EXERCISES 5. Using a graphing calculator, find the mean and standard deviation of the following frequency distribution. Find the percentage of scores that lie within 1 and 2 standard deviations of the mean. Explain why this is or is not a normal distribution. Scores 5 6 7 8 9 10 11 6. Frequency 9 11 22 38 19 13 8 The administration of a medical clinic wishes to find out how long people have to wait to be seen by a doctor. The waiting times (in minutes) for 20 patients were recorded as follows: 5.5 7.9 4.5 9.5 7.9 4.2 5.8 1.5 12.0 15.0 4.8 3.3 1.5 6.5 10.6 12.7 8.8 11.0 20.0 7.8 Find the mean, the median and the standard deviation for this data. Is this data considered a normal distribution for one standard deviation? DAKOTA COLLEGIATE Page 5 of 10 Variability and Statistical Analysis SENIOR 4 APPLIED MATHEMATICS EXERCISES Exercise 3: Z-scores 1, The average mark in Sarah’s English class is 60%, with a standard deviation of 16. The average mark in her math class is 58%, with a standard deviation of 10. If Sarah scores 72% in English, and 68% in math, how many standard deviations is each of her grades above average? What does this tell us about her performance in the two subjects? 2. Peter’s goal is to maintain his marks at least 2.5 z-scores above the mean in all of his subject. Determine the minimum marks he must obtain in each subject. Subject Mean Chemistry English Math 66 62 68 Standard Deviation 7 12 8 Physics 73 4 Minimum Mark 3. Two students applying for a scholarship have the following scores. Johanna has a mark of 82% in the College Entrance Survey Test, which has a mean of 75% and standard deviation of 8%. Harvey has a mark of 78% in the Scholarship Aptitude Exam, which has a mean of 70% and standard deviation of 5%. Use standardized scores to compare the marks. 4. A college conducts an entrance exam consisting of an English language skills test and a mathematics test. Alycia scored 210 out of 300 on the language test and 540 out of 600 on the mathematics test. Compare Alycia’s performance on the two parts of the test in the following cases. English Test Mean Case 1 Case 2 Case 3 5. 80% 60% 60% Mathematics Test Standard Deviation 5% 10% 5% Mean 83% 83% 90% Standard Deviation 7% 3.5% 3.5% The IdleWyld College of Advanced Technology advertises that the minimum requirement in mathematics for entry to courses in the Computer Engineering Department is 90% in an Entrance Test in which the mean mark is known to be 78% with standard deviation of 8%. Liem has taken the Joint Examining Boards test (which is acceptable to the College that has a mean of 73% with standard deviation of 10%. What is the minimum acceptable mark in this test? DAKOTA COLLEGIATE Page 6 of 10 SENIOR 4 APPLIED MATHEMATICS EXERCISES Variability and Statistical Analysis Exercise 4: Normal distribution and Z-Scores Use the graphing calculator to solve the following problems. 1. In a normal distribution, find the probability that a score picked at random lies in each interval described by the following z-scores. a) z< 0 d) z < 3 g) z > 2.5 b) z<1 e) z < 3.5 h) z > -2 c) z < 2.5 f) z > 0 2. In a normal distribution, find the probability that a score picked at random lies in each interval described by the following z-scores. a) –1 < z <1 d) -1.645< z < 1.645 g) -0.5 < z < 1.2 b) –1.96 < z < 1.96 e) 1 < z < 2 c) –2 < z < 2 f) 0.5 < z <2.7 3. Find the z-score, p. where p > 0, so that the area between z = 0 and z = p is given. a) Area = 0.1700 b) Area = 0.4500 c) Area = 0.4750 d) Area = 0.49400 4. Find the z-score, p and -p so that the area between z = -p and z = p is given. a) Area = 0.9000 e) Area = 0.7500 b) Area = 0.5000 d) Area = 0.9500 5. In a normal distribution, find the probability that a score picked at random lies in each interval described by the following 2-scores. a) z >1.96 or z< -1.96 b) z > 2.5 or z < -2.5 c) z > 1 or z < -1 d) z > 0.675 or z < -0.675 6. According to Nielsen Media Research, people watch television an average of 6.98 hours per day. Assume that those times are normally distributed with a standard deviation of 3.80 hours. Find percentage of viewers who watch television more than 8.0 hours per day 7. In a court of law a woman testified that she gave birth 300 days after conception. Normal pregnancies last 268 days with a standard deviation of 15 days. Is it reasonable to believe that she could be telling the truth? Explain your reasoning. DAKOTA COLLEGIATE Page 7 of 10 SENIOR 4 APPLIED MATHEMATICS EXERCISES Variability and Statistical Analysis 8. The weights of men aged 18 to 74 are normally distributed with a mean of 173 pounds and a standard deviation of 30 pounds (based on a national health survey). Find the percentage of the population that has weights between 190 and 225 pounds. In a group of 400 men aged 18 to 74 years, how many are expected to weigh between 190 and 225 pounds? 9. Scores on a university entrance biology aptitude test have a mean score of 8.0 and a standard deviation of 2.6. If 600 prospective students wrote the test, how many would be expected to score between 6.0 and 7.0? 10. The cholesterol levels in men aged 18 to 74 are normally distributed with a mean of 178,1 and a standard deviation of 40.7. All units are in mg/1OO mL of blood. What is the probability that the cholesterol level of a randomly selected man (18 to 74 years) is between 100 and 200? 11. The heights of six-year-old girls are normally distributed with a mean of 117.8cm and a standard deviation of 5.52 cm. Find the probability that the height of a randomly selected six-year-old girl will be between 117.8 cm and 120.56 cm. 12. The mean pulse rate of males aged 18 to 25 years is 72 beats per minute, and the standard deviation is 97. If the military states that a rate of 100 or higher is unsuitable for army recruits, what portion of the male population (18 to 25) would be unsuitable for military service on these grounds? 13. Two students have obtained the following marks in comparable math courses. Juana has 82% in an exam with a mean score of 78% and a standard deviation of 5%. Hans scored 73% in an exam with a mean of62% and standard deviation of 8%. Use z-scores to compare their results. Comment on these results. 14. The principal and staff of a school decide to use z-scores to translate test marks into letter grades. They agree to the following: F -D -C -B -A– less than –1 from –1 and less than –0.5 from –0.5 and less than 0.5 from 0.5 and less than 1 1 and above. The test scores for one class of 36 students were: 23, 34, 36, 39, 42, 44, 48, 50, 52, 54, 54, 55, 62, 62, 63, 64, 64, 65, 66, 67, 70, 71, 71, 75, 80, 81, 83, 85, 87, 88, 89, 94, 96, 98 100, 100 Find the zones of scores for the letter grades. DAKOTA COLLEGIATE Page 8 of 10 SENIOR 4 APPLIED MATHEMATICS EXERCISES Variability and Statistical Analysis Exercise 5: Confidence Intervals (For each of the following find the mean and the standard deviation) np npq 1. The survey of all the students in a school indicated that 32% of the students are left-handed. Find an interval for the number of left-handed students expected to be enrolled in a class of 24 students 95% of the time. 2. Records at a garden center indicate that 75% of a particular kind of tulip bulb will grow successfully. A customer buys 20 such bulbs. Find the 95% interval for the number of bulbs from the 20 that will grow successfully. 3. Fred takes a duck hunting holiday. He has a probability of 0.4 of hitting any duck that he shoots. He plans to take 30 shots. Find a confidence interval for the number of ducks he could expect to shoot that would be true 19 times out of 20. 4. If 15% of all computer chips manufactured are faulty, find a 90% confidence interval for the mean number of faulty chips in batches of 100 chips. 5. A long term study shows that 20% of all cars have oil leaks. A sample of 140 cars is tested for oil leaks. Would it be reasonable to expect 40 cars to have oil leaks? Explain. 6. A shoe store owner knows from past experience that 75% of all customers pay with credit cards. In one day the store sold shoes to 120 customers. How many customers would you expect to pay by credit card 19 times out of 20? 7. A recruiter for the armed forces finds that 55 out of 200 candidates tested are unfit for service. Find a 95% confidence interval for the actual percentage of candidates who are unfit for service. 8. The highway patrol discover that, at one location, there is a 60% probability that a motorist will exceed the speed limit. Assume that 3500 cars drive this road daily. Determine a 90% confidence interval for the mean number of motorists exceeding the speed limit daily 9. In a sample of 100 students, 72 favoured a change in school hours. Find a 95% confidence interval for the percentage of all students in favour of a change in school hours. DAKOTA COLLEGIATE Page 9 of 10 SENIOR 4 APPLIED MATHEMATICS EXERCISES Variability and Statistical Analysis Exercise 6: Opinion Polls (For each of the following find the mean and the standard deviation) np npq 1. There are two candidates in an election, A and B. A poll of 400 voters selected at random finds that 208 intend to vote for Candidate A. Give a 95% confidence interval for the percentage of voters favourable to A at the time of the poll. (Assume that there are no undecided voters.) 2. A public opinion survey has shown that 80 persons in a random sample of 400 persons answered “Yes” to a certain question. Within what limits can it be asserted that the true population percentage of “Yes” voters would lie? 3. In a random sample of 100 people in a particular district, 70 people were in favour of declaring March 2lst a public holiday. Find a 95% confidence limit for the percentage of all people who would be in favour of this public holiday. 4. A pollster interviewed 1000 people selected randomly and found that 650 of them preferred Brand X to Brand Y. Find a 95% confidence interval for the percentage of the whole population that would prefer Brand X. DAKOTA COLLEGIATE Page 10 of 10