Sample Test # 3
Sample Test # 3 Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. ____ ____ 1. The heights of men, in centimetres, are normally distributed with a mean of 175 and a standard deviation of 20. If Mario is 180 cm tall, what percent of men is he taller than? a. 0.62% c. 59.87% b. 40.13% d. 99.38% 2. The masses of bolts made in a plant are normally distributed. Bolts will be rejected if their z-scores are ____ 3. ____ 4. ____ 5. ____ 6. ____ 7. ____ greater than 2.15 or less than 2.10. What percent of bolts will be rejected? a. 0% c. 1.79% b. 1.58% d. 3.37% The heights, in centimetres, of the 700 female students in a high school are normally distributed with a mean of 158 and a standard deviation of 6. How many of these students have a height between 151 cm and 165 cm? a. 527 c. 530 b. 528 d. 531 The men that shop for clothes at the Hard-to-Fit Shoppe are typically unusually short or unusually tall. The distribution of their heights is likely to be a. U-shaped c. mound-shaped b. uniform d. skewed The distributions of men’s masses and women’s masses are both mound-shaped. But since men tend to have greater masses than women, the distribution of the masses of all adults will likely be a. bimodal c. right-skewed b. assymetric d. left-skewed How many of the numbers in the set below are within two standard deviations of the mean? 0, 2, 4, 6, 7, 8 a. 3 c. 6 b. 4 d. 8 Which of the following measures of spread could be negative? a. the range c. the standard deviation b. the IQR d. none of the above 8. If X~N(12.4, ) and 95% of the data lie in the interval 11.8–13.0 the equals a. 1.2 c. 0.09 b. 0.3 d. 0.06 ____ 9. The variable has a normal distribution with 99.7% of the area under its curve falling symmetrically between x = 50 and x = 170. Its mean and standard deviation are respectively a. 110 and 20 c. 120 and 20 b. 110 and 202 d. 120 and 202 ____ 10. A university accepts only applicants scoring in the top 16% on an entrance test. Each year the test scores are normally distributed with a standard deviation of 30. What is the highest value that the mean can have for Fred to be accepted with a score of 520? a. 460 c. 520 b. 490 d. 550 Short Answer 11. Calculate the z-score, to one decimal place, of x = 7.2 if = 8.1 and = 3. 12. Suppose X~N(50, 42 ). What value of x would have a z-score of 2.10? 13. The maximum attention spans, in seconds, of 40 two-year-olds are normally distributed with a mean of 75 and a standard deviation of 10. How many of these toddlers will have an attention span of less than 71 s? 14. How many peaks are there in a perfectly U-shaped distribution? 15. What type of distribution has an assymetrical histogram? 16. “Some of you did well, some of you did not do so well, but most of you have a mark near the middle.” What shape of distribution would these marks likely have? 17. Find the range of the following data set: 2, 7, 0, 1, 5, 9 18. What is the interquartile range for the following average January temperatures (in /C): 2.0, 1.5, 0.2, 1.7, 2.6, 4.1, 4.1, 5.0, 7.2, 8.1, 8.1 19. The variance of a set of data is 13.2. Find the standard deviation to one decimal place. 20. Find the standard deviation, to one decimal place, of the IQ scores listed below: IQ Frequency 80 3 90 4 100 8 110 3 120 2 21. Use the standard deviation to determine which bowling scores were more consistent and state the standard deviations of those scores to the nearest decimal place. Onorio Marianne 22. X~N(12.2, 170 161 165 172 181 148 174 163 149 178 ) and 99.7% of data fall within the interval 11.6–12.8. What is the standard deviation? 23. An internet site has an average of 2500 hits per day. The number of daily hits is approximately normally distributed with a standard deviation of 220. For how many of the next 50 days would you expect the site to receive fewer than 2280 hits? 24. A traffic study showed that vehicle speeds on a particular highway were normally distributed with a mean of 102.5 km/h and a standard deviation of 5.1 km/h. What percent of vehicles had a speed between 92.3 km/h and 107.6 km/h? Problem 25. What is the approximate range of a set of data that has been displayed in a histogram with 10 bars each with width 0.5? 26. The distribution of the ages of people at a Punctured Lung concert was mound-shaped with eighteen-year-olds representing the largest age group in attendance. Fred attended the concert with his ten year old son. Why did he not see many people near either of their ages at the Lung concert? 27. Why is the mode, but not the mean or median an appropriate measure of central tendency for qualitative data? 28. A set of seven numbers is arranged from smallest to largest. The IQR is zero and the third number is 14. Is it possible to determine the median? Explain. 29. The masses of largemouth bass, in kilograms, are normally distributed with a mean of 1.1 and a standard deviation of 0.3. Any bass caught with a mass between 0.8 kg and 1.7 kg must be released. What percent of the bass does this represent? 30. Suppose X~N(2, 32 ). Which set of data contains a greater percent of the data? a) x 1 together with x 31. For X~N( , 5 or b) 2 x 5? ), approximately 44% of the data are less than 2.9. Find the value of , to one decimal place. 32. Why do normal z-score tables typically not include z values greater than 2.99? 33. What are the mean and standard deviation of the standard normal distribution? 34. How many of the 250 perch netted by a fisherman would you expect to have a mass more than 400 g, if the distribution of their masses in approximately normal with a mean of 300 g and a standard deviation of 80 g? 35. The masses of Burgerville’s half-pounders are normally distributed. Of these burgers, 33% have masses greater then 253.52 g and 40.9% of them have masses less than 248.16 g. Find the mean and standard deviation of the half-pounder masses. 36. Outliers are data values much larger or smaller than most of the other data values. Where would any intervals containing outliers occur for each of these distributions: U-shaped, uniform, mound-shaped, right-skewed, leftskewed? 37. The noon-day temperature (in /C) in Chillsville last winter can be represented by the normal distribution X~N( 6, 32 ). How often was it above 0/C at noon in Chillsville last winter? 38. Construct a suitable histogram for the following temperatures (in /C) on the TI-83+. 1, 0, 1, 1, 2, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 10, 11 39. Explain how measures of spread are used to measure consistency. 40. If each number in a data set is multiplied by a constant m, then the standard deviation should also be multiplied by m. Verify this statement for the data sets below: x: 0, 2, 4, 10 y: 0, 6, 12, 30 41. State the mean, standard deviation, and variance if X~N(4, 32 ). Sample Test # 3 Answer Section MULTIPLE CHOICE 1. ANS: OBJ: TOP: 2. ANS: OBJ: TOP: 3. ANS: LOC: 4. ANS: OBJ: TOP: 5. ANS: OBJ: TOP: 6. ANS: LOC: 7. ANS: LOC: 8. ANS: LOC: 9. ANS: LOC: 10. ANS: LOC: C REF: Knowledge and Understanding 3.5 Applying the Normal Distribution: Z-Scores LOC: ST3.03 Tools for Analyzing Data D REF: Knowledge and Understanding 3.5 Applying the Normal Distribution: Z-Scores LOC: ST3.03 Tools for Analyzing Data D REF: Application OBJ: 3.5 Applying the Normal Distribution: Z-Scores ST3.03 TOP: Tools for Analyzing Data A REF: Knowledge and Understanding 3.1 Graphical Displays of Information LOC: ST3.01 Tools for Analyzing Data A REF: Knowledge and Understanding 3.1 Graphical Displays of Information LOC: ST3.01 Tools for Analyzing Data C REF: Knowledge and Understanding OBJ: 3.3 Measures of Spread ST2.01 TOP: Tools for Analyzing Data D REF: Application OBJ: 3.3 Measures of Spread ST2.02 TOP: Tools for Analyzing Data B REF: Knowledge and Understanding OBJ: 3.4 Normal Distribution ST3.02 TOP: Tools for Analyzing Data A REF: Application OBJ: 3.4 Normal Distribution ST3.02 TOP: Tools for Analyzing Data B REF: Application OBJ: 3.4 Normal Distribution ST3.02 TOP: Tools for Analyzing Data SHORT ANSWER 11. ANS: The z-score is 0.3. REF: Knowledge and Understanding OBJ: 3.5 Applying the Normal Distribution: Z-Scores LOC: ST2.03 TOP: Tools for Analyzing Data 12. ANS: The value of x would be 58.4. REF: Knowledge and Understanding OBJ: 3.5 Applying the Normal Distribution: Z-Scores LOC: ST3.03 TOP: Tools for Analyzing Data 13. ANS: Of these toddlers, 14 will have an attention span of less than 71 s. REF: Application OBJ: 3.5 Applying the Normal Distribution: Z-Scores LOC: ST3.03 TOP: Tools for Analyzing Data 14. ANS: The number of peaks in a perfectly U-shaped distribution is 2. REF: Knowledge and Understanding OBJ: 3.1 Graphical Displays of Information LOC: ST3.01 TOP: Tools for Analyzing Data 15. ANS: A skewed distribution has an assymetrical histogram. REF: Knowledge and Understanding OBJ: 3.1 Graphical Displays of Information LOC: ST3.01 TOP: Tools for Analyzing Data 16. ANS: These marks would likely have a distribution that has a mound-shape. REF: Knowledge and Understanding OBJ: 3.1 Graphical Displays of Information LOC: ST3.01 TOP: Tools for Analyzing Data 17. ANS: The range is 10. REF: Knowledge and Understanding OBJ: 3.3 Measures of Spread LOC: ST2.01 TOP: Tools for Analyzing Data 18. ANS: The average January temperature is 7.0ºC. REF: Knowledge and Understanding OBJ: 3.3 Measures of Spread LOC: ST2.01 TOP: Tools for Analyzing Data 19. ANS: The standard deviation is 3.6. REF: Knowledge and Understanding OBJ: 3.3 Measures of Spread LOC: ST2.01 TOP: Tools for Analyzing Data 20. ANS: The standard deviation is 11.5. REF: Knowledge and Understanding OBJ: 3.3 Measures of Spread LOC: ST2.01 TOP: Tools for Analyzing Data 21. ANS: Marianne’s bowling scores were more consistent. The standard deviation for her scores is 10.2. REF: Knowledge and Understanding OBJ: 3.3 Measures of Spread LOC: ST2.02 TOP: Tools for Analyzing Data 22. ANS: The standard deviation is 0.2. REF: Knowledge and Understanding OBJ: 3.4 Normal Distribution LOC: ST3.02 TOP: Tools for Analyzing Data 23. ANS: The site would receive fewer than 2280 hits on 8 of the next 50 days. REF: Application OBJ: 3.4 Normal Distribution LOC: ST3.02 TOP: Tools for Analyzing Data 24. ANS: The percent of vehicles with a speed between 91.3 km/h and 107.6 km/h is 81.5%. REF: TOP: Application OBJ: 3.4 Normal Distribution Tools for Analyzing Data LOC: ST3.02 PROBLEM 25. ANS: Bin width = range 0.5 = range number intervals 10 Therefore, the range is 5. REF: Knowledge and Understanding OBJ: Chapter 3 Prob LOC: ST3.01 TOP: Tools for Analyzing Data 26. ANS: Mound-shaped distributions have decreased frequencies on either side of the interval with greatest frequency. So there would be few people much younger or older than 18 at the concert. REF: Communication OBJ: Chapter 3 Prob LOC: ST3.01 TOP: Tools for Analyzing Data 27. ANS: The mean and median need the data to have properties of numbers to be evaluated, for example, the capacity to be added and divided, and order. The mode can be evaluated with any type of data since only frequency is required. REF: Communication OBJ: Chapter 3 Prob LOC: ST2.01 TOP: Tools for Analyzing Data 28. ANS: Yes, it is possible to determine the median. IQR = Q3 =0 Q1 Q1 = Q3 Since the numbers are arranged in increasing order, this means that all of the numbers between Q1 and Q3, inclusive, are equal. But Q1 is the second number and Q3 is the sixth number, so this includes the third number and the fourth number which is the median. Therefore, the median equals 14. REF: Thinking/Inquiry/PS OBJ: Chapter 3 Prob LOC: ST2.01 TOP: Tools for Analyzing Data 29. ANS: 0.8 is one standard deviation below the mean and 1.7 is two standard deviations above the mean. Therefore, the percent released is 68% + 13.5% = 81.5%. REF: Application OBJ: Chapter 3 Prob LOC: ST3.02 TOP: Tools for Analyzing Data 30. ANS: The first set consists of data not within one standard deviation of the mean, so it contains 32% of the data. The second set consists of the data within one standard deviation above the mean, and it contains 34% of the data. Therefore, the set in b) contains a greater percent of the data. REF: Knowledge and Understanding OBJ: Chapter 3 Prob LOC: ST3.02 TOP: Tools for Analyzing Data 31. ANS: If P(Z < z) 0.44, then z = 0.15 REF: Knowledge and Understanding OBJ: Chapter 3 Prob LOC: ST3.03 TOP: Tools for Analyzing Data 32. ANS: Almost all of the data in a normal distribution are within three standard deviations of the mean. Thus, data with z-scores of three or greater represent a negligible proportion of the total. REF: Communication OBJ: Chapter 3 Prob LOC: ST3.03 TOP: Tools for Analyzing Data 33. ANS: The mean = 0. The standard deviation = 1. REF: Knowledge and Understanding OBJ: 3.5 Applying the Normal Distribution: Z-Scores LOC: ST3.02 TOP: Tools for Analyzing Data 34. ANS: Therefore, 26 fish should have a mass more than 400 g. REF: Application LOC: ST3.03 35. ANS: For Z~N(0, 1): OBJ: 3.5 Applying the Normal Distribution: Z-Scores TOP: Tools for Analyzing Data If , then And If so , then . . So and Solving this linear system gives g and g. REF: Thinking/Inquiry/PS OBJ: 3.5 Applying the Normal Distribution: Z-Scores LOC: ST3.03 TOP: Tools for Analyzing Data 36. ANS: The intervals would appear on either the extreme right or extreme left side and have a small frequency, assuming that the bin width is not too great. Therefore, for U-shaped and uniform distributions there would not be outlier intervals. For mound-shaped distributions, the intervals containing outliers would occur on both the left and right sides of the distribution. For right-skewed distributions, the intervals containing outliers would be on the right side and for left-skewed distributions they would be on the left side. REF: Thinking/Inquiry/PS OBJ: 3.1 Graphical Displays of Information LOC: ST3.01 TOP: Tools for Analyzing Data 37. ANS: x > 0 = –6 + 2(3), so a positive temperature is more than 2 standard deviations above the mean. Therefore, this would have occured on 2.5% of the days. REF: Application OBJ: 3.4 Normal Distribution TOP: Tools for Analyzing Data 38. ANS: LOC: ST3.02 REF: Application OBJ: 3.1 Graphical Displays of Information LOC: STV.02 TOP: Tools for Analyzing Data 39. ANS: If a set of data has a small measure of spread, then the data are relatively close to each other and therefore consistent. Conversely, if the measure of spread is large, then the data are more dispersed and therefore not as consistent. REF: Communication OBJ: 3.3 Measures of Spread LOC: ST2.02 TOP: Tools for Analyzing Data 40. ANS: m=3 and and REF: Knowledge and Understanding OBJ: 3.3 Measures of Spread LOC: ST2.01 TOP: Tools for Analyzing Data 41. ANS: The mean = 4. The standard deviation = 3. The variance = 9. REF: Knowledge and Understanding OBJ: 3.4 Normal Distribution LOC: ST3.02 TOP: Tools for Analyzing Data
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