Mt. Douglas Secondary

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1 Foundations of Math 11 Calculator Usage 207 HOW TO USE TI-83, TI-83 PLUS, TI-84 PLUS CALCULATORS FOR STATISTICS CALCULATIONS shows it is an actual calculator key to press 1. Using LISTS to Calculate Mean, Median and Standard Deviation A. To clear existing values from L1 and L2: STAT (move cursor down to ClrList) ENTER 2nd L1, 2nd L2 ENTER ClrList L1, L2 Done B. To enter new values in L1 AND L2: STAT (Edit) ENTER (now use cursor key to enter data values in L1 and frequencies in L2) L1 L C. To calculate mean, median and standard deviation: STAT (move cursor to CALC) ENTER 2nd L1, 2nd L2 ENTER 1 Var Stats L1 L2 D. Read mean x = Standard deviation σ x = (cursor down 3 times) Median Med = 1 Var Stats x = σ x = Med = 2. To Calculate the Area under a Normal Curve A. Method 1 using normalcdf 2nd DISTR (move cursor to 2: normalcdf) ENTER Syntax: normalcdf (Z a,z b ) area between Z a and Z b normalcdf (a, b, μ, σ) area between a and b use 1E99 for left extreme (or 99) and 1E99 for right extreme (or 99) e.g., normalcdf ( 1E99, b, μ, σ) area to left of b normalcdf (a, 1E99, μ, σ) area to right of a B. Method 2 using ShadeNorm ( Z a, Z b ) i) Set WINDOW values: X min = 3 X min = μ 3 σ = X max = 3 X max = μ + 3 σ = Y max = 0.4 X max = 0.4 σ = Y min = 0.1 Y min = (Y max 4) allows room for text

2 208 Calculator Usage Foundations of Math 11 ii) Clear previous normal curve: 2nd DRAW (ClrDraw) ENTER ENTER iii) Draw the normal curve: 2nd DISTR (move cursor to DRAW ) ENTER ShadeNorm ( Z a, Z b ), ClrDraw Done iv) Once you have finished recording results, ClrDraw again: See step (ii) above. C. Method 3 using ShadeNorm (a, b, μ, σ) Example: find the probability that a person has an IQ < 120 given the mean IQ is 100, with standard derviation 15. i) CALCULATE WINDOW VALUES, (using μ and σ) μ = 100, σ = 15 X min = μ 3σ = = 55 X max = μ + 3σ = = 145 X max = 0.4 σ = Y min = (Y max 4) = (0.03) ClrDraw Done ii) Clear previous normal curve: See step (ii) in Method II. iii) Draw the normal curve: 2nd DISTR DRAW ENTER ShadeNorm ( 1E99, 120, 100, 15) ENTER ClrDraw Done, Syntax: ShadeNorm (a, b, μ, σ) use a = 1E99 for left extreme area to left of b use b = 1E99 for right extreme area to right of a Note: You can also use 99 and 99 for the left and right extremes.

3 Foundations of Math 11 Calculator Usage To Calculate Z-score for a Standard Normal Distribution A. Method 1 Using invnorm (area) x u σ 2nd = Z Z = invnorm (area) DISTR (move cursor to) invnorm ENTER (enter value) ENTER Note: the area is for a Normal Curve. B. Method 2 Using invnorm (area, μ, σ ) 2nd DISTR (move cursor to) invnorm ENTER (enter values) ENTER Enter z-score area, then μ = mean, σ = standard deviation Note: the area is for a Standard Normal Curve. Note: area is zero at 1E99 and one at 1E99, therefore, area varies from left to right between zero and one.

4 210 Chapter 5 Statistics Foundations of Math 11

5 Foundations of Math 11 Section 5.1 Mean, Median and Mode Mean, Median and Mode Statistics is a field of Mathematics dealing with the collecting and summarizing of data. Once the data has been gathered, the information is evaluated and analyzed so that a decision based on these measurable events can be made. Society in general depends to a great extent on our ability to evaluate information, determine what is true, and make correct decisions. It is amusing to remember the famous words of Mark Twain when interpreting information, There are three kinds of lies lies, damn lies, and statistics. Mean, Median and Mode Mean, median and mode are all measures of central tendency or average. These represent, respectively, average, centre and most frequent of the data gathered. Mean The mean is computed by adding a set of values and dividing by the total number of values. x = mean of a sample, μ = mean of the whole population. i) μ = x 1 + x 2 + x x n n ii) μ = iii) μ = n x i i=1 n n i=1 n f i x i basic mean formula mean using sigma notation mean using group distribution f i = frequency Note: the sum of n numbers with a mean of μ is S n = n μ. Example 1 Determine the mean of: 1, 6, 3, 8, 9, 3 Solution: μ = = 5

6 212 Chapter 5 Statistics Foundations of Math 11 Example 2 For 30 randomly selected high school students, the following IQ frequency distribution was obtained. Determine the mean. Class Limits Frequency 80 x < x < x < x < x < x < Solution: Method 1 Midpoint of class limits are 85, 95, 105, 115, 125, 135 x = Method 2 (by TI-83 Calculator) = STAT (EDIT) ENTER (enter values under L1 & L2) STAT (move cursor to CALC) ENTER (1 Var Stats) 2nd L1, 2nd L2 ENTER Mean Example 3 Solution: 10 numbers have a mean of 37. If one number is removed, the mean is 38. What was the number that was removed? Sum of 10 numbers: S 10 = = 370 Sum of 9 numbers: S 9 = 9 38 = 342 Removed number was: = 28

7 Foundations of Math 11 Section 5.1 Mean, Median and Mode 213 Median The median is the middle value in an odd number of values. In an even number of values, the median is the mean of the two middle values. Before one can find this point, the data must be arranged in order. By formula, the median is the n +1 term. 2 Example 1 Odd number of values: Solution: Median = terms, so the median is = 3rd term Example 2 Even number of values: } Median = = 5.5 Solution: 6 terms, so the median is = 3.5, therefore, median = 3rd term + 4 th term 2 Example 3 For 30 randomly selected high school students, the following IQ frequency distribution was obtained. Determine the median. Class Limits Frequency 80 x < x < x < x < x < x < 140 1

8 214 Chapter 5 Statistics Foundations of Math 11 Solution: Method 1 By formula, the median term is n = = So, the median is 15th term + 16 th term. 2 Both the 15 th term and 16 th term are found in the interval 100 x < 110 so the median is 105. Method 2 (by TI-83 Calculator) STAT (EDIT) ENTER (enter values under L1 & L2) STAT (move cursor to CALC) ENTER (1 Var Stats) 2nd L1, 2nd L2 ENTER (move cursor down to Med) Median Mode The mode is the value that occurs most often in a set of values. If all values are equal in frequency, then there is no mode; there can be multiple modes if same values occur equally often. Example 1 What is the mode of the following data set? 0, 0, 1, 1, 1, 2, 2, 2, 3, 3 Solution: Both 1 and 2 have a frequency of 3, therefore the modes are 1 and 2. Example 2 For 30 randomly selected high school students, the following IQ frequency distribution was obtained. Determine the mode. Solution: Class Limits Frequency 80 x < x < x < x < x < x < The range 100 x < 110 has frequency of 11, so the mode is 105.

9 Foundations of Math 11 Section 5.1 Mean, Median and Mode Exercise Set 1. The value of the middle term in a ranked data set is called the a) mean b) median c) mode 2. Which of the following is affected by extreme values? a) mean b) median c) mode 3. Which of the following can have more than one value? a) mean b) median c) mode 4. Determine the mean, median and mode for the following set of values: 1, 2, 3, 4, 4, 7 5. The incomes of a sample of 6 local teachers are as follows: $41 500, $44 900, $39 700, $62 300, $ and $ What is the mean, median and mode income of the 6 teachers? 6. Determine the mean, median and mode salaries of the staff listed below: Staff Salary One owner $ One Manager $ Two salespersons $ Six technicians $44 000

10 216 Chapter 5 Statistics Foundations of Math The following frequency distributions represents the monthly commission in dollars for 25 car salespersons at a car lot. Determine the mean, median and mode. 8. The following table gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a publishing company. Commission in $ Frequency 800 x < x < x < x < Number of Orders Number of Days Calculate the mean, median and mode. 9. The mean age of five people is 39. The ages of four of these five persons are 33, 45, 27 and 41. Find the age of the fifth person. 10. The mean score of 18 female students on a math test is 72 and the mean score of 14 male students is 66. Find the combined mean score.

11 Foundations of Math 11 Section 5.1 Mean, Median and Mode A small business has 10 people in total on the payroll. 8 workers make $ per year 1 foreman makes $ per year 1 owner makes $ per year a) Determine the mean, median and mode for the 10 people. 12. a) What form of central tendency is always part of the data? b) What form of central tendency requires using all values of the data? c) What form of central tendency is the centre or middle value of the data? b) If you were the owner, what type of average would you prefer to use in wage bargaining? Why? d) What form of central tendency is least likely to be an actual score of the data? c) If you were a worker, what type of average would you prefer to use in wage bargaining? Why? e) Which of the three values of central tendency can assume more than one value for the data? 13. Consider the data sets Set I: Set II: Notice that each value of the second data set is obtained by adding 6 to the corresponding value of the first data set. Calculate the mean of the two data sets, and comment on the relationship between the two means. 14. Consider the data sets Set I: Set II: Notice that each value of data set II is obtained by multiplying the corresponding value of the first data set by 2. Calculate the mean of these data sets, and comment on the relationship between the two means.

12 218 Chapter 5 Statistics Foundations of Math If there are 8 numbers with a mean of 10 and 12 other numbers with a mean of 16, what is the mean of all 20 numbers? 16. If the mean of 50 numbers is 18 and the mean of the first 30 numbers is 16, what is the mean of the last 20 numbers? 17. The mean of three numbers is If the 3 second number is one less than 3 times the first, and the third is one more than twice the second, what are the 3 numbers? 18. The mean of 50 numbers is 38. If two of the numbers, namely 45 and 55, are removed, what is the mean of the remaining numbers?

13 Foundations of Math 11 Section 5.2 Standard Deviation Standard Deviation Consider the following question: 30 students are in the same Math and English classes; the mean for a test in Math and a test in English is 50%. If Mari scored 60% in Math and 70% in English, in which class did she do better compared to other students in the two classes? The answer to this question is that you cannot tell because you don t know the spread or dispersion of test scores about the mean in the two classes. The value that measures the spread or dispersion of data about the mean is called standard deviation. The standard deviation is based on the deviations from the mean. We square the difference between each value and the mean. This squaring eliminates negative numbers. We then total the squared deviations, and divide the total by the number of values. The square root of this value is the standard deviation Standard Deviation Definition σ = Greek symbol (sigma) for standard deviation of population. σ = sum of the squares of the differences from the mean number of values = (x 1 μ)2 + (x 2 μ) (x n μ) 2 n (basic formula) n = 1 (x n i μ) 2 (summation notation) i=1 n = 1 x 2 n i μ 2 (alternate form of summation notation) i=1

14 220 Chapter 5 Statistics Foundations of Math 11 Example 1 Calculate the standard deviation from the following sets of values: a) 5, 6, 7, 8, 9 b) 3, 5, 7, 9, 11 Solution: Method 1 (by formula) 1. a) First find the mean: μ = = 7 σ = 1 5 [(5 7)2 + (6 7) 2 + (7 7) 2 + (8 7) 2 + (9 7) 2 ] or = 1 ( ) 5 = 2 σ = 1 5 ( ) 7 2 = 1 ( ) 49 5 = = 2 1. b) First find the mean: μ = = 7 σ = 1 5 [(3 7)2 + (5 7) 2 + (7 7) 2 + (9 7) 2 + (11 7) 2 ] or = 1 ( ) 5 = 2 2 σ = 1 5 ( ) 7 2 = 1 ( ) 49 5 = = 2 2 Notice that (3, 5, 7, 9, 11) are two units apart compared to one unit apart for (5, 6, 7, 8, 9). So the standard deviation is twice as large.

15 Foundations of Math 11 Section 5.2 Standard Deviation 221 Method 2 (by TI-83 Calculator) 2. a) STAT (EDIT) ENTER (enter values under L1) STAT (move cursor to CALC, then l : 1 Var Stats) ENTER ENTER Standard Deviation 2. b) STAT (EDIT) ENTER (enter values under L1) STAT (move cursor to CALC, then 1 Var Stats) ENTER ENTER Standard Deviation Note: A small standard deviation indicates the measures are clustered very close to the mean and a high standard deviation indicates the measures are widely scattered from the mean.

16 222 Chapter 5 Statistics Foundations of Math 11 Example 2 Calculate the standard deviation for the following sets of data: a) Daily Commute Time (minutes) Number of Employees 0 to less than to less than to less than to less than to less than b) Number of Orders Number of Days Solution: Method 1 1. a) First find the mean: μ = σ = or σ = 5(4) +15(9) + 25(6) + 35(4) + 45(2) 25 = = (5 21.4) 2 + 9( ) 2 + 6( ) 2 + 4( ) 2 + 2( ) 2 = = b) First find the mean: μ = 11(4) +14(12) +17(20) + 20(14) 50 = = σ = 1 50 [ ] or = σ = 4( )2 +12( ) ( ) 2 +14( ) 2 50 =

17 Foundations of Math 11 Section 5.2 Standard Deviation 223 Method 2 2. a) STAT (EDIT) ENTER (enter values under L1, L2) STAT (move cursor to CALC, then 1 Var Stats L1, L2) ENTER ENTER Mean Standard Deviation 2. b) STAT (H:ClrList) ENTER (ClrList L1, L2) ENTER (Done) STAT (EDIT) ENTER STAT (move cursor to CALC, then 1 Var Stats L1, L2) ENTER ENTER Mean Standard Deviation

18 224 Chapter 5 Statistics Foundations of Math Exercise Set 1. The value of standard deviation is a) never negative b) always positive c) never zero 2. Find the standard deviation of 2, 3, 5, Find the standard deviation. a) Score Frequency b) Score Frequency 0 x < x < x < x < Find the standard deviation of the following graph: What do each of the following signify? a) small standard deviation b) large standard deviation c) standard deviation of zero

19 Foundations of Math 11 Section 5.2 Standard Deviation All the required calculations to find the mean and standard deviation appear below: Number of Orders f (frequency) m (midpoint) mf m x (m x) 2 f (m x ) a) What is the mean? b) What is the standard deviation?

20 226 Chapter 5 Statistics Foundations of Math Fill in the table, and then find the standard deviation of 25 TVs sold during the past 25 weeks. Number of Orders f (frequency) m (midpoint) mf m x (m x) 2 f (m x ) 2 8. Without doing any calculations, what is the relationship between the standard deviation of 1, 2, 3, 4, 5 and the standard deviation of a) 3, 5, 7, 9, Each of the following lists contains just one value which is different from all the others. Without calculating, what relationship is the standard deviation between the values? Set I: Set II: b) 501, 502, 503, 504, 505 c) 21, 24, 27, 30, 33

21 Foundations of Math 11 Section 5.2 Standard Deviation Each of the following lists contain an even number of values of only two different values. Without calculating, relate the standard deviation to the difference between the values. Set I: Set II: Set III: The following scores are the results of the midterm test given to five classes of 100 Calculus students at the University of Saskatchewan. A score of 40 is needed to pass, and a score of 85 is needed to get an A. Group Mean Standard Deviation Which of the five groups most likely a) shows the highest student score? b) shows the lowest student score? c) shows the most uniform scores? d) has the biggest range of scores? e) has the most failures? f) has the most A s? g) represents the overall average of all 5 classes?

22 228 Chapter 5 Statistics Foundations of Math The Normal Distribution If you gathered data from experiments such as the height, weight and test scores of individuals, it would follow a set pattern. This pattern, called a normal distribution, is extremely important because you only need the mean and standard deviation to complete the entire distribution. Important characteristics of the Normal Curve μ It is bell shaped and symmetric about the mean. The enclosed area always equals one. The probability that an event happens equals the area under the normal curve. The standard deviation is related to the area under the curve. The curve will never touch the x-axis but extends to infinity in both directions. The mean, median and mode are always the same. Two normal curves with the same standard deviation, but different means. x1 x2 Small standard deviation Large standard deviation Two normal curves with the same mean, but different standard deviations. x

23 Foundations of Math 11 Section 5.3 The Normal Distribution 229 Z-score of Standard Normal Curve There are many different possible normal curves with different values of μ and σ. By transforming each score into a z-score, which is a measure of how far a value is from the mean, the z-scores will fit the standard normal curve with μ = 0 and σ = 1. Definition Z = difference between x and μ standard deviation = x μ σ Z: number of standard deviations that x is away from the mean μ μ: mean x : a particular score σ : standard deviation Note that there are just as many negative z-scores (when x < μ) as there are positive z-scores. Standard Normal Curve The value under the curve indicates that approximate proportion of area in each section. 3 σ 2 σ 1 σ μ = 0 1 σ 2 σ 3 σ Note that over 68% of the data are within 1 standard deviation of the mean, and over 95% of the data are within 2 standard deviations. Relating Normal Curve and Standard Normal Curve P(a X b) = P a μ σ < Z < b μ σ μ Z = a μ a σ Z = b μ b σ

24 230 Chapter 5 Statistics Foundations of Math 11 Example 1 If IQ scores are normally distributed with a mean of 100 and standard deviation of 15, determine a) the z-score for 120 b) the probability that a randomly selected person has an IQ less than 120 Solution: a) Z = x μ σ = = b) P( X 120) = P Z < = P(Z < 4 3 ) Method 1 (by table at back of chapter) Find z-table in text, the table value always indicates area or probability to the LEFT of the z-score, therefore, P(Z < 1.33) = This score indicates that 90.82% of a normal population have an IQ less than 120. Method 2 (by TI-83 calculator) i) 2nd DISTR (move cursor down to normalcdf) ENTER (enter values) ENTER Or normalcdf(left extreme, Z b ) normalcdf(left extreme, b, μ, σ) Note: 1E99 indicates a lower bound value of the standard normal curve. ii) WINDOW (enter values) 2nd DISTR (move cursor to DRAW) ENTER (enter values) ShadeNorm ( 1E99, 4/3) ENTER Note: try to memorize these max/min values for Z curve ShadeNorm(left extreme, Z b )

25 Foundations of Math 11 Section 5.3 The Normal Distribution 231 iii) WINDOW (enter values) 2nd DISTR (move cursor to DRAW) ENTER (enter values) ShadeNorm ( 1E99, 120, 100, 15) ENTER ShadeNorm(left extreme, b, μ, σ) Note: If doing by ShadeNorm method, be sure to clear graph when finished by entering 2nd DRAW (ClrDraw) ENTER (DONE) Example 2 The grade point average at Penticton Secondary is 2.6, with a standard deviation of 0.5. If the top 10% of all students are eligible to attend U.B.C., what is the minimum G.P.A. needed to attend U.B.C.? (Top 10% means that 90% will be less than the minimum G.P.A.) Solution: P( X 2.6) = P Z < x = 0.9 x = Inverse Z 0.9 Method 1 (by table) Go to the Standard Normal Distribution Table and locate the closest value to 0.9 which is Once the closest number is found, read the value in the left column and top row to find the z-score value. The z-score for is Therefore, x 2.6 = 1.28 x = So, a G.P.A. of 3.24 is needed to go to U.B.C. 0.5 Method 2 (by TI-83 Calculator) i) 2nd DISTR (move cursor to invnorm) ENTER (enter values) ENTER Or invnorm (area) x 2.6 = 1.28 x = So, a G.P.A. of 3.24 is needed to go to U.B.C. invnorm (area, μ, σ )

26 232 Chapter 5 Statistics Foundations of Math Exercise Set 1. Find the area on the standard normal curve between z and z if a) z = 1 2. Decide what area under the standard normal curve is bigger or if the two areas are equal. a) the area between z = 1 and z = 1 or the area between z = 0 and z = 2 b) z = 2 b) the area between z = 0.2 and z = 0.3 or the area between z = 1.2 and z = 1.3 c) z = 3 c) the area between z = 1 and z = 0.5 or the area between z = 0.5 and z = 1 d) z = 4 d) the area between z = 1 and z = 2 or the area between z = 2 and z = 4 e) z = 5 e) the area to the right of z = 2.5 or the area to the right of z = 1.5

27 Foundations of Math 11 Section 5.3 The Normal Distribution Find the area under the standard normal curve a) between z = 0.62 and z = Find the area under a normal distribution curve with μ = 4andσ = 10 a) area between x = 3 and x = 9 b) between z = 2.35 and z = 1.42 b) area from x = 0 to x = 15 c) between z = 1.42 and z = 2.38 c) area to the right of x = 6 d) to the right of z = 1.46 d) area to the left of x = 2.31 e) to the right of z = 2.37 e) area to the right of x = If the probability of z < a = 0.8, determine a. 6. If the probability of x b = 0.6 and μ = 4, σ = 10, determine b.

28 234 Chapter 5 Statistics Foundations of Math The attendance for a week at the local theatre is normally distributed, with a mean of 4000 and a standard deviation of 500. What percent of attendance figures fall between 3600 and 4600 people? 8. Women pay on average $600 more for a car than men. Assume a normal distribution of charging with a mean of $600 and a standard deviation of $50. Find the probability that a woman pays at least $675 more than a man for a car. 9. A manufacturer of cell phones indicated a mean of 26 months before there is a need of repairs, with a standard deviation of 6 months. What length of time for the warranty should the manufacturer set such that less than 10% of all cell phones will need repairs during the warranty period? 10. A provincial math exam has a mean of 68 and a standard deviation of If students take the exam, and a score of 49 or less fails, how many students fail the exam?

29 Foundations of Math 11 Section 5.3 The Normal Distribution A normal random variable has a standard deviation of 4. If the probability that X is less than 15 is 0.75, what is the mean of X? 12. A normal random variable X has a mean of 80. If the probability that X is less than 72 is 15%, what is the standard deviation of X? 13. A test to a large group of students approximates the normal curve. The mean is 70 with a standard deviation of 8. If 8% of the students receive A s, and 16% receive B s, what is the minimum mark needed to receive a B? 14. At a high school, the average grade for English is 64, with a standard deviation of 10. If 20 students with grades between 73 and 85 receive B s, how many students are taking English at the high school?

30 236 Chapter 5 Statistics Foundations of Math Confidence Interval for Means The concept of confidence intervals has to be one of the most important theories in all of statistics. A person often hears statements like: 1 of 5 Canadians is currently dieting the life expectancy of smokers is five years less than non-smokers the average teacher makes $ a year with results accurate to within two points, 19 times out of 20. Not every person in Canada is surveyed to obtain these results. One must then ask how many people were in the survey, and how accurate are the results. Sample measures are used to estimate population measures. Confidence interval is defined as a specific interval estimate of the whole population by using information obtained from a sample, and the specific confidence level of the estimate. It may be reasoned that the mean of the sample x should be centered about the mean of the whole population μ. Because x is a single number, it is called a point estimate of the population mean μ. When the point estimate x is complete, the accuracy of the estimate is not really known. Statisticians have developed a method of taking a sample, and the standard normal distribution curve, to make an accurate estimate of the whole population. STANDARD NORMAL DISTRIBUTION left tail = 2 α E 1 α E x right tail = 2 α Z α = μ E 2 μ Z α = μ + E 2 For a given α, there is a probability of 1 α that x will miss μ by less than the value of E = maximum error. The confidence interval is defined as: Confidence Level Theorem x Z α 2 σ n < μ < x + Z σ α 2 n x E < μ < x + E for some population with maximum error E = Z α 2 where σ n μ = population mean x = sample mean Z α = standard deviation z score separated into 2 tails 2 n = sample size σ = s = standard deviation of sample Note: A sample size n > 30 must be used in this formula because σ, the standard deviation of the whole population and s, the standard deviation of a sample, are very close when n > 30.

31 Foundations of Math 11 Section 5.4 Confidence Interval for Means 237 Example 1 A random sample of size 64 has a mean of 160 and a standard deviation of 15. Find a 95% confidence interval for the true mean of the population. Note: Z = 1.96 from table or invnorm (0.025) or invnorm (0.975) Solution: Method 1 (by formula) x Z α 2 σ n < μ < x + Z α 2 σ n (1.96) 64 < μ < (1.96) < μ < < μ < Method 2 (by TI-83 calculator) STAT (move cursor to TESTS, & down to ZInterval) (enter values) ENTER (move cursor to Calculate) ENTER Which you read as < μ < Hence, you can say with 95% confidence that the true mean of the whole population is between and Or, as a reporter will say, the true mean of the population is 160, within 4, 19 times out of 20.

32 238 Chapter 5 Statistics Foundations of Math Exercise Set 1. The mean of the sampling distribution of x is always equal to a) μ 2. The standard deviation of the sampling distribution of the sample mean decreases when a) x increases b) Z α 2 b) n increases c) σ n c) n decreases 3. When samples are selected from a normally distributed population, the sampling distribution of the sample mean has a normal distribution a) if n When samples are selected from a nonnormally distributed population, the sampling distribution of the sample mean has a normal distribution a) if n 30 b) if n 100 b) if n 100 c) all the time c) all the time

33 Foundations of Math 11 Section 5.4 Confidence Interval for Means Determine Z α for the following to two decimal places: 2 a) Z α 2 for 99% confidence interval b) Z α 2 for 98% confidence interval c) Z α 2 for 95% confidence interval d) Z α 2 for 90% confidence interval e) Z α 2 for 80% confidence interval

34 240 Chapter 5 Statistics Foundations of Math Determine the 95% confidence interval for μ if σ = 6, x = 72 and n = Determine the 80% confidence interval for μ if σ = 6, x = 72 and n = Determine the 95% confidence interval for μ if σ = 6, x = 72 and n = Determine the 80% confidence interval for μ if σ = 6, x = 72 and n = 49.

35 Foundations of Math 11 Section 5.4 Confidence Interval for Means A study of 50 English teachers found the average time spent marking a term paper was 15.2 minutes with a standard deviation of 2.8 minutes. Find a 94% confidence interval of the mean time for all term papers. 11. Forest companies bid on a large tract of land in the Prince George forest district. A random sample of 150 trees yields a mean diameter of 48 cm with a standard deviation of 5.6 cm. Find a 90% confidence interval for the mean diameter of all the trees. 12. A real-estate firm in Winnipeg takes a random sample of 60 homes. This sample yields a mean of 1800 square feet of living space with a standard deviation of 280 square feet. Construct a 99% confidence level for the mean square footage of living space for all Winnipeg homes. 13. Find the sample size necessary to estimate a population mean to within 4 units if σ = 15. We want 90% confidence level in our results.

36 242 Chapter 5 Statistics Foundations of Math A study of 50 people living in Crescent Beach, BC, showed the average age as 42 years with a standard deviation of 12 years. a) Find the 95% confidence interval of the mean age for all the people living in Crescent Beach. 15. A random sample of 400 passengers that arrive at Vancouver International Airport has a mean processing time of 45 minutes with a standard deviation of 12 minutes. a) Construct a 98% confidence interval for the mean arrival time for all passengers. b) If the 95% confidence interval of the study stays the same, but we have 100 people instead of 50, what happens to the confidence interval? Why? b) If the range you obtained in part a) is larger than you want, how can you narrow it? 16. The confidence interval < μ < is found by using a random sample for which x = 80.2, σ = 5.3 and n = 40. Determine the degree of confidence. 17. A high school teacher wishes to estimate the number of hours a student spends studying each week. The standard deviation from a previous study was 2.5 hours. How large a sample must be selected if the teacher wants to be 98% confident that the true mean differs from the sample mean by 0.75 hours?

37 Foundations of Math 11 Section 5.5 Chapter Review Chapter Review 1. A population of scores is normally distributed with a mean of 62.4 and a standard deviation of If 40% of the scores are higher than a particular score x, calculate the value of x. a) 59.3 b) 65.5 c) 69.6 d) Determine the value of a in P(Z > a) = a) b) c) d) Determine the value of a in P(a < Z < 0) = a) 2.73 b) 1.24 c) 1.24 d) 2.73

38 244 Chapter 5 Statistics Foundations of Math Let Z be a random variable with standard normal distribution. If P(a Z 2) = 0.1, determine a. a) b) c) d) Which frequency distribution shows a set of outcomes with the largest standard deviation? a) b) c) d) 6. Determine the mean and standard deviation of the following graph. a) μ = 2.87, σ = 1.50 b) μ = 2.87, σ = 1.58 c) μ = 3, σ = 1.41 d) μ = 3, σ = 1.50

39 Foundations of Math 11 Section 5.5 Chapter Review 245 Use the following information to answer questions 7 and 8. Bed Day offers savings of $10, $25, $50 and $100 off on their already low price on bedroom suites. Select a coupon and save the amount on the coupon. There are a total of 100 coupons with 70, 20, 8 and 2 containing $10, $25, $50 and $100, respectively. 7. Determine the average saving. a) $18.00 b) $25.00 c) $31.25 d) $ Determine the standard deviation. a) b) c) d) 39.45

40 246 Chapter 5 Statistics Foundations of Math 11 Use the following information to answer questions 9 and 10. Let X be normally distributed with mean 6 and standard deviation Determine P(5 X 10). a) b) c) d) Determine b if P( X b) = a) 3.98 b) 7.04 c) 8.02 d) How does the mean of 9x, 10x, 11x, 12x compare to the mean of 3x, 4x, 5x, 6x? a) 2 times as large b) 2 1 times as large 3 c) 3 times as large d) 6 units larger

41 Foundations of Math 11 Section 5.5 Chapter Review How does the standard deviation of 4x, 6x, 8x, 10x compare to the standard deviation of x, 2x, 3x, 4x? a) the same b) 2 times as large c) 4 times as large d) 8 times as large 13. Determine the standard deviation of a fair die. a) b) c) 3.5 d) A teacher assigns grades in Mathematics according to the following procedure: A if score exceeds μ + 1.3σ B if score between μ + 0.4σ and μ + 1.3σ C if score between μ 0.5σ and μ + 0.4σ D if score between μ 1.5σ and μ 0.5σ E if score is below μ 1.5σ If there are 30 students in the class, how many students receive C s, assuming the scores are normally distributed? a) 8 b) 10 c) 12 d) 14

42 248 Chapter 5 Statistics Foundations of Math A set of 500 scores are normally distributed. How many scores would you expect to find between 0.8 standard deviations and 2.5 standard deviations above the mean? a) 100 b) 101 c) 102 d) A four-point grading system gives 4 points for an A, 3 points for a B, 2 1 points for a C+, 2 points for a C, 2 and 1 point for a pass, with no points for a fail. If a student s grades consist of 8 A s, 15 B s, 6 C+ s, 8 C s, 2 passes and 1 fail, what is his grade point average? a) 2.08 b) 2.50 c) 2.75 d) The mean diameter of Okanagan apples is 12.5 cm with a standard deviation of 1.2 cm. If the diameter is normally distributed, what is the minimum diameter needed to only reject 10% of all apples? a) cm b) cm c) cm d) cm

43 Foundations of Math 11 Section 5.5 Chapter Review From the list below, determine the minimum number of students with scores of 90 required to have the class average over 75. Score Number of Students 7 3 8? a) 5 b) 6 c) 7 d) Find z if the standard normal curve area between z and z is a) ± b) ± c) ± d) ± The average height of adult males is 70 inches with a standard deviation of 2.4 inches. If the heights are normally distributed, how high should a doorway be such that 95% of all men can pass through the doorway without hitting their heads. a) 71 inches b) 72 inches c) 73 inches d) 74 inches

44 250 Chapter 5 Statistics Foundations of Math If a random variable has the normal distribution with μ = and σ = 5.7, find the probability that the value will be greater than a) 0.08 b) 0.09 c) 0.10 d) In test A, a student received a mark of 43 with class mean 30 and standard deviation 10. In test B, a student received a mark of 70 with class mean 60 and standard deviation 8. In test C, a student received a mark of 75 with class mean 60 and standard deviation In test D, a student received a mark of 54 with class mean 50 and standard deviation 3. In which test did the student get the highest standard score? a) A b) B c) C d) D 23. The number of complaints about food in the school cafeteria per month is a random variable having the normal distribution with μ = 8.3 and σ = 1.8, find the probability that in any month exactly 10 complaints will be reported. a) 0.12 b) 0.13 c) 0.14 d) 0.15

45 Foundations of Math 11 Section 5.5 Chapter Review In order to be a candidate for the R.C.M.P, recruits are given a stress test. The scores are normally distributed with a mean of 62 and a standard deviation of 8.4. If just the top 25% of recruits are selected, determine the minimum score needed. a) 66 b) 67 c) 68 d) If the weight of female high school students closely follows the normal distribution with a mean of 125 pounds and a standard deviation of 5 pounds, what range of weights includes the middle 80% of the girls in high school? a) to b) to c) to d) to In Math 11, the average grade is 64, and the standard deviation is 10. The teacher s distribution has 8 students from 60.0 to 67.0 receiving C s. Assuming normal distribution of grades, how many students are in the Math class? a) 27 b) 29 c) 31 d) 33

46 252 Chapter 5 Statistics Foundations of Math Determine the standard deviation of the population a 2d, a d, a, a + d, a + 2d. a) 2d b) 3d c) d d) 2d 28. Most shoes last 2.4 years with a standard deviation of 1.2 years. A shoe company guarantees shoes for 1 year with free replacement. For every 500 shoes sold, assuming normal distribution, how many pairs of shoes must have free replacements? a) 58 b) 59 c) 60 d) One of Canada s top universities only accepts the top 12% of all high school graduates on the basis of exam results. The exam results are normally distributed with a mean of 500 and a standard of 100. Determine the minimum score needed to enter this university. a) 614 b) 616 c) 618 d) 620

47 Foundations of Math 11 Section 5.5 Chapter Review A random variable has a normal distribution with σ = 3.0. If the probability is that this random variable will take on a value less than 85.2, what is the probability that it takes on a value greater than 78.4? a) b) c) d)

48 254 Chapter 5 Statistics Foundations of Math 11 THE STANDARD NORMAL DISTRIBUTION TABLE F z Area 0 z ( z)= P[ Z < z] z

49 Foundations of Math 11 Section 5.5 Chapter Review 255 F z ( z)= P[ Z < z] z

50 256 Chapter 5 Statistics Foundations of Math 11