Assignment No 2 ndash Normal Distribution

z

1 X is a normally normally distributed variable with mean μ = 30 and standard deviation σ = 4 Find a) P(x lt 40) b) P(x gt 21) c) P(30 lt x lt 35)

2 A radar unit is used to measure speeds of cars on a motorway The speeds are normally distributed with a mean of 90 kmhr and a standard deviation of 10 kmhr What is the probability that a car picked at random is travelling at more than 100 kmhr

3 For a certain type of computers the length of time bewteen charges of the battery is normally distributed with a mean of 50 hours and a standard deviation of 15 hours John owns one of these computers and wants to know the probability that the length of time will be between 50 and 70 hours

4 Entry to a certain University is determined by a national test The scores on this test are normally distributed with a mean of 500 and a standard deviation of 100 Tom wants to be admitted to this university and he knows that he must score better than at least 70 of the students who took the test Tom takes the test and scores 585 Will he be admitted to this university

5 The length of similar components produced by a company are approximated by a normal distribution model with a mean of 5 cm and a standard deviation of 002 cm If a component is chosen at random a) what is the probability that the length of this component is between 498 and 502 cm b) what is the probability that the length of this component is between 496 and 504 cm

6 The length of life of an instrument produced by a machine has a normal ditribution with a mean of 12 months and standard deviation of 2 months Find the probability that an instrument produced by this machine will last a) less than 7 months b) between 7 and 12 months

7 The time taken to assemble a car in a certain plant is a random variable having a normal distribution of 20 hours and a standard deviation of 2 hours What is the probability that a car can be assembled at this plant in a period of time a) less than 195 hours b) between 20 and 22 hours

8 A large group of students took a test in Physics and the final grades have a mean of 70 and a standard deviation of 10 If we can approximate the distribution of these grades by a normal distribution what percent of the students a) scored higher than 80 b) should pass the test (gradesge60) c) should fail the test (gradeslt60)

9 The annual salaries of employees in a large company are approximateley normally distributed with a mean of $50000 and a standard deviation of $20000 a) What percent of people earn less than $40000 b) What percent of people earn between $45000 and $65000 c) What percent of people earn more than $70000

10 An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days Assuming that bulb life is normally distributed what is the probability that an Acme light bulb will last at most 365 days

11 Suppose scores on an IQ test are normally distributed If the test has a mean of 100 and a standard deviation of 10 what is the probability that a person who takes the test will score between 90 and 110

12 Molly earned a score of 940 on a national achievement test The mean test score was 850 with a standard deviation of 100 What proportion of students had a higher score than Molly (Assume that test scores are normally distributed)

13 13 It was found that the mean length of 100 parts produced by a lathe was 2005 mm with a standard deviation of 002 mm Find the probability that a part selected at random would have a length

Assignment No 2 ndash Normal Distribution

(a) between 2003 mm and 2008 mm

(b) between 2006 mm and 2007 mm

(c) less than 2001 mm

(d) greater than 2009 mm

14 A company pays its employees an average wage of $325 an hour with a standard deviation of 60 cents If the wages are approximately normally distributed determine

a the proportion of the workers getting wages between $275 and $369 an hourb the minimum wage of the highest 5

15 The average life of a certain type of motor is 10 years with a standard deviation of 2 years If the manufacturer is willing to replace only 3 of the motors that fail how long a guarantee should he offer Assume that the lives of the motors follow a normal distribution

16 If X is a normal random variable with mean (m) 96 and standard deviation (σ) 10 Calculate P(Xlt106)

17Calculate the following Normal distribution probabilities

(a) P(Z gt 106)

(b) P(Z lt -215)

(c) P(106 lt Z lt 400)

(d) P(-106 lt Z lt 400)

18 Let us now consider the problem of making probability statements about a variable that is known to have a standard normal distribution Let us suppose that we wish to calculate the following probabilities

(a) That Z is greater than 24 ie p( z gt 24)

(b) That Z lies between -124 and 186 ie p(-124 lt Zlt 186)

19 The Tahoe Natural Coffee Shop morning customer load follows a normal distribution with mean 45 and standard deviation 8 Determine the probability that the number of customers tomorrow will be less than 42

20 A study was done to determine the stress levels that students have while taking exams The stress level was found to be normally distributed with a mean stress level of 82 and a standard deviation of 134 What is the probability that at your next exam you will have a stress level between 9 and 10

ExampleThe IQrsquos of a group students are normally distributed with a mean of 100 and a standard deviation of 12 What is the lowest IQ of the top 30 of the students Call this IQ xFirst draw a simple diagram The areamarked by lines will need to be 30To find the value of Z for which the shaded area is 30 we need to look at the numbersin the standard Normal tableThe nearest to 30 is 03015 which is along the row Z = 05 and under the columnZ = 002 so Z = 052Z = (value given-mean)standard deviation052 = ( x- 100 ) 12

Assignment No 2 ndash Normal Distribution

12 X 052 = x - 100624 = x - 100x = 10624The lowest IQ of the top 30 of the students is 106 (to 3 figures)Exercise1 The blood pressure of adult males is normally distributed with a mean of 125mmhg and a standard deviation of 8a) What is the minimum blood pressure ofthe 50 with the highest blood pressuresb) What is the minimum blood pressure of the 30 with the highest blood pressuresc) What is the maximum blood pressure of the 30 with the lowest blood pressuresd) What is the minimum blood pressure of the 20 with the highest blood pressuresAnswers (a) 125mmHg (b) 12916mmHg(c) 12084mmHg (d) 1318mmHgExampleThe weights of a packet of biscuits are normally distributed with a standard deviation of 05gThe packing machine can be set to any given mean weightWhat must the mean be set at if 85 ofpackets must weigh more than 500gFirst draw the diagram

The shaded area is 85 so the unshaded tail area is 15This is the area we look up in the table to find the value of ZThe nearest to 15 is 01492 for whichZ = -104 (negative because the area is on the left)Z = (value given-mean)standard deviation-104 = (500 - M) 05- 104 X 05 = 500 - M- 052 = 500 - MM = 500 + 052 = 50052gThe mean should be set at 50052gExercise1 What should the mean be set at in the previous example if 80 of packets of biscuits must weigh more than 500g2 A machine packs small stones into bags for use in gardens The standard deviation is 05kg The mean fill of the bags can be set to any value What should it be set at so that 70 of bags weigh at least 26kg3 The marks of a group of students are normally distributed with a standard deviation of 10 If 42 of the students gained 67 marks or more out of 100 what was the mean mark4 It is known that the salaries of employees at ABC plc are normally distributed with a standard deviation of pound10 000 If 70 of employees have a salary of at least pound20 000 what is the mean salary of all employesAnswers 1) 50042g 2) 2574kg 3) 65 4) pound25 200

5 In a statistics test the marks were normally distributed with a mean of 68 and a standard deviation of 10a) What was the minimum mark of the top 45 of studentsb) The top 15 of students are to receive a special prize What is the least mark for a prizewinnerc) What mark did the bottom 20 of students fall below6 The time taken to complete a test is normally distributed with a mean of 5 minutes and standard deviation of 1 minutea) How long did the slowest 15 take to complete the testb) How quick were the quickest 377 The salaries of workers at Hardywork plc are normally distributed with a mean of pound20 000 and standard deviation of pound5 000 What is the minimum salary of the highest earning 70Answers 5a) 69 (b) 78 (c) 60 6a) 604 mins (b) 467 mins 7 pound17 400

• 19 The Tahoe Natural Coffee Shop morning customer load follows a normal distribution with mean 45 and standard deviation 8 Determine the probability that the number of customers tomorrow will be less than 42
• 20 A study was done to determine the stress levels that students have while taking exams The stress level was found to be normally distributed with a mean stress level of 82 and a standard deviation of 134 What is the probability that at your next exam you will have a stress level between 9 and 10

Assignment No 2 ndash Normal Distribution

(a) between 2003 mm and 2008 mm

(b) between 2006 mm and 2007 mm

(c) less than 2001 mm

(d) greater than 2009 mm

14 A company pays its employees an average wage of $325 an hour with a standard deviation of 60 cents If the wages are approximately normally distributed determine

a the proportion of the workers getting wages between $275 and $369 an hourb the minimum wage of the highest 5

15 The average life of a certain type of motor is 10 years with a standard deviation of 2 years If the manufacturer is willing to replace only 3 of the motors that fail how long a guarantee should he offer Assume that the lives of the motors follow a normal distribution

16 If X is a normal random variable with mean (m) 96 and standard deviation (σ) 10 Calculate P(Xlt106)

17Calculate the following Normal distribution probabilities

(a) P(Z gt 106)

(b) P(Z lt -215)

(c) P(106 lt Z lt 400)

(d) P(-106 lt Z lt 400)

18 Let us now consider the problem of making probability statements about a variable that is known to have a standard normal distribution Let us suppose that we wish to calculate the following probabilities

(a) That Z is greater than 24 ie p( z gt 24)

(b) That Z lies between -124 and 186 ie p(-124 lt Zlt 186)

19 The Tahoe Natural Coffee Shop morning customer load follows a normal distribution with mean 45 and standard deviation 8 Determine the probability that the number of customers tomorrow will be less than 42

20 A study was done to determine the stress levels that students have while taking exams The stress level was found to be normally distributed with a mean stress level of 82 and a standard deviation of 134 What is the probability that at your next exam you will have a stress level between 9 and 10

ExampleThe IQrsquos of a group students are normally distributed with a mean of 100 and a standard deviation of 12 What is the lowest IQ of the top 30 of the students Call this IQ xFirst draw a simple diagram The areamarked by lines will need to be 30To find the value of Z for which the shaded area is 30 we need to look at the numbersin the standard Normal tableThe nearest to 30 is 03015 which is along the row Z = 05 and under the columnZ = 002 so Z = 052Z = (value given-mean)standard deviation052 = ( x- 100 ) 12

Assignment No 2 ndash Normal Distribution

12 X 052 = x - 100624 = x - 100x = 10624The lowest IQ of the top 30 of the students is 106 (to 3 figures)Exercise1 The blood pressure of adult males is normally distributed with a mean of 125mmhg and a standard deviation of 8a) What is the minimum blood pressure ofthe 50 with the highest blood pressuresb) What is the minimum blood pressure of the 30 with the highest blood pressuresc) What is the maximum blood pressure of the 30 with the lowest blood pressuresd) What is the minimum blood pressure of the 20 with the highest blood pressuresAnswers (a) 125mmHg (b) 12916mmHg(c) 12084mmHg (d) 1318mmHgExampleThe weights of a packet of biscuits are normally distributed with a standard deviation of 05gThe packing machine can be set to any given mean weightWhat must the mean be set at if 85 ofpackets must weigh more than 500gFirst draw the diagram

The shaded area is 85 so the unshaded tail area is 15This is the area we look up in the table to find the value of ZThe nearest to 15 is 01492 for whichZ = -104 (negative because the area is on the left)Z = (value given-mean)standard deviation-104 = (500 - M) 05- 104 X 05 = 500 - M- 052 = 500 - MM = 500 + 052 = 50052gThe mean should be set at 50052gExercise1 What should the mean be set at in the previous example if 80 of packets of biscuits must weigh more than 500g2 A machine packs small stones into bags for use in gardens The standard deviation is 05kg The mean fill of the bags can be set to any value What should it be set at so that 70 of bags weigh at least 26kg3 The marks of a group of students are normally distributed with a standard deviation of 10 If 42 of the students gained 67 marks or more out of 100 what was the mean mark4 It is known that the salaries of employees at ABC plc are normally distributed with a standard deviation of pound10 000 If 70 of employees have a salary of at least pound20 000 what is the mean salary of all employesAnswers 1) 50042g 2) 2574kg 3) 65 4) pound25 200

5 In a statistics test the marks were normally distributed with a mean of 68 and a standard deviation of 10a) What was the minimum mark of the top 45 of studentsb) The top 15 of students are to receive a special prize What is the least mark for a prizewinnerc) What mark did the bottom 20 of students fall below6 The time taken to complete a test is normally distributed with a mean of 5 minutes and standard deviation of 1 minutea) How long did the slowest 15 take to complete the testb) How quick were the quickest 377 The salaries of workers at Hardywork plc are normally distributed with a mean of pound20 000 and standard deviation of pound5 000 What is the minimum salary of the highest earning 70Answers 5a) 69 (b) 78 (c) 60 6a) 604 mins (b) 467 mins 7 pound17 400

• 19 The Tahoe Natural Coffee Shop morning customer load follows a normal distribution with mean 45 and standard deviation 8 Determine the probability that the number of customers tomorrow will be less than 42
• 20 A study was done to determine the stress levels that students have while taking exams The stress level was found to be normally distributed with a mean stress level of 82 and a standard deviation of 134 What is the probability that at your next exam you will have a stress level between 9 and 10

Assignment No 2 ndash Normal Distribution

12 X 052 = x - 100624 = x - 100x = 10624The lowest IQ of the top 30 of the students is 106 (to 3 figures)Exercise1 The blood pressure of adult males is normally distributed with a mean of 125mmhg and a standard deviation of 8a) What is the minimum blood pressure ofthe 50 with the highest blood pressuresb) What is the minimum blood pressure of the 30 with the highest blood pressuresc) What is the maximum blood pressure of the 30 with the lowest blood pressuresd) What is the minimum blood pressure of the 20 with the highest blood pressuresAnswers (a) 125mmHg (b) 12916mmHg(c) 12084mmHg (d) 1318mmHgExampleThe weights of a packet of biscuits are normally distributed with a standard deviation of 05gThe packing machine can be set to any given mean weightWhat must the mean be set at if 85 ofpackets must weigh more than 500gFirst draw the diagram

The shaded area is 85 so the unshaded tail area is 15This is the area we look up in the table to find the value of ZThe nearest to 15 is 01492 for whichZ = -104 (negative because the area is on the left)Z = (value given-mean)standard deviation-104 = (500 - M) 05- 104 X 05 = 500 - M- 052 = 500 - MM = 500 + 052 = 50052gThe mean should be set at 50052gExercise1 What should the mean be set at in the previous example if 80 of packets of biscuits must weigh more than 500g2 A machine packs small stones into bags for use in gardens The standard deviation is 05kg The mean fill of the bags can be set to any value What should it be set at so that 70 of bags weigh at least 26kg3 The marks of a group of students are normally distributed with a standard deviation of 10 If 42 of the students gained 67 marks or more out of 100 what was the mean mark4 It is known that the salaries of employees at ABC plc are normally distributed with a standard deviation of pound10 000 If 70 of employees have a salary of at least pound20 000 what is the mean salary of all employesAnswers 1) 50042g 2) 2574kg 3) 65 4) pound25 200

5 In a statistics test the marks were normally distributed with a mean of 68 and a standard deviation of 10a) What was the minimum mark of the top 45 of studentsb) The top 15 of students are to receive a special prize What is the least mark for a prizewinnerc) What mark did the bottom 20 of students fall below6 The time taken to complete a test is normally distributed with a mean of 5 minutes and standard deviation of 1 minutea) How long did the slowest 15 take to complete the testb) How quick were the quickest 377 The salaries of workers at Hardywork plc are normally distributed with a mean of pound20 000 and standard deviation of pound5 000 What is the minimum salary of the highest earning 70Answers 5a) 69 (b) 78 (c) 60 6a) 604 mins (b) 467 mins 7 pound17 400

• 19 The Tahoe Natural Coffee Shop morning customer load follows a normal distribution with mean 45 and standard deviation 8 Determine the probability that the number of customers tomorrow will be less than 42
• 20 A study was done to determine the stress levels that students have while taking exams The stress level was found to be normally distributed with a mean stress level of 82 and a standard deviation of 134 What is the probability that at your next exam you will have a stress level between 9 and 10