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QUESTION 4 Pulse rates of adult men are approximately normally distributed with a mean of 70 and a standard deviation of 8. Which of the following choices correctly describes how to determine the proportion of men that have a pulse rate greater than 787 (A) Determine the area to the left of z= 1 under the standard normal curve. (B) Determine the area to the right of z = 1 under the standard normal curve. (C) Determine the area to the right of z = —1 under the standard normal curve. (D) Determine the area between z =—1 and z = 1 under the standard normal curve.
QUESTION 7 The life expectancy (in years) of an electronic component can be represented by the probability density function 1 p(x) =142 0 otherwise > 1 The probability that the component lasts between 1 and 10 years is (A) 0.010 (B) 0.100 (C) 0.900 (D) 0.990
QUESTION 14 (6 marks) Let X denote the time in minutes between the arrival of trains at a station. The cumulative distribution function of X is defined by —— 5<x<10 F(x)= X 0 otherwise a) Determine the probability density function of X. [3 marks] b) Determine the probability that 5 < X <7. [1 mark] c¢) Determine the mean time between the arrival of trains at the station. [2 marks]
QUESTION 2 The table shows the time a technician has spent servicing photocopiers. What is the probability that a given service required at least 10 minutes but less than 20 minutes? (A) 0.15 (B) 0.35 (C) 0.70 (D) 0.85
QUESTION 8 The continuous random variable X has the probability density function - 1<x< -~ —_— = S Il = 2 b | W 0 otherwise The mean of X 1s (A) ln(%) it ©) h{%) (D) 1
QUESTION 10 Handspans of teenagers are approximately normally distributed with a mean of 15 cm and a standard deviation of 2 cm. Which of the following groups is expected to be the largest? (A) teenagers with handspans that are between 7 cm and 11 cm (B) teenagers with handspans that are between 11 cm and 15 cm (C) teenagers with handspans that are between 13 cm and 17 cm (D) teenagers with handspans that are between 17 cm and 21 cm
QUESTION 13 (7 marks) The amount of gravel (in tonnes) sold by a construction company in a given week is a continuous random variable X and has a probability density function defined by: f`(x)={c(l_x2)* 0<x<l 0 otherwise 3 a) Show that ¢ = 5 b) Determine P(X <0.25). c¢) Determine the variance of X. [1 mark] [2 marks] [4 marks]
QUESTION 14 (7 marks) The heights of students at School A are normally distributed with a mean of 165 cm and a standard deviation of 15 cm. a) Determine the probability that a student chosen at random from School A is shorter than 180 cm. [1 mark] b) Determine the minimum integer value of the height of a student who is in the top 2% of this distribution. [3 marks] The heights of students at School B are also normally distributed. A student at School B has the same height as the height determined in Question 14b) but their corresponding z-score is 3. c) Determine which student`s height ranks higher in terms of percentile for their school. [3 marks]
QUESTION 5 The continuous random variable X has the probability density function cos(x) —rx T 48 < & f(x)={ 2 > 2772 0o otherwise The standard deviation of X is (A) 0.467 (B) 0.684 (C) 1.211 (D) 1.467
QUESTION 7 The distribution of marks in three subjects is given below. The mark received by a student in each of the subjects is also shown. Science 60 15 70 Mathematics 68 6 72 Music 65 9 72 Standardised z-scores were used to compare the results and showed that the student (A) performed equally well in mathematics and science. (B) performed equally well in mathematics and music. (C) performed better in mathematics than in science. (D) performed better in mathematics than in music.
QUESTION 17 (7 marks) A continuous random variable X has a probability density function f defined by: _(a+bx?*? 0<x<1 ! (x)—{ 0 otherwise 3 i IfE(X) = = determine a and b.
QUESTION 7 Determine the mean of the continuous random variable X with the probability density function 1 fO)=48 0 otherwise O<x<4 (A) (B) (C) (D) WloO wplRe WlWw CwleH
QUESTION 10 The continuous random variable Y has the probability density function l+y 0<y<¥ 3-1 f(y) = 0 otherwise ] Determine P(O< y< 5 (ay 1 5 (iB) 3 8 (Cc) > 8 3 D) = (D) 1
QUESTION 3 In a certain normal distribution curve 95% of the area lies between the values 50.32 and 113.68. The mean of this distribution is 82. Determine the standard deviation. (A) 16.16 (B) 21.12 (C) 31.68 (D) 63.36
QUESTION 10 A student is trying to determine which subject they performed best in compared to other students. Results from recent tests in four subjects (A to D) are shown. Assume student results in each subject are normally distributed. In which subject did the student perform best compared to other students? Class standard Class mean a deviation Student’s result
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