Chapter 6 Normal Distributions

Chapter 6 Normal DistributionsWhat is Normal Distribution?

Characteristics of Normal DistributionLets test your knowledge

Do these distributions have the same mean?

Compare the two graphs, what can you tell me about the standard deviation?AnswerA) Both graphs have the same mean

B) Red curve has a smaller standard deviation because the data is more compressed. Black curve has a bigger standard deviation because the data is more spread out.Empirical ruleFor a distribution that is symmetrical and bell-shaped (normal distribution):Approximately 68% of the data values will lie within one standard deviation on each side of the meanApproximately 95% of the data values will lie within two standard deviations on each side of the meanApproximately 99.7% of the data values will lie within three standard deviations on each side of the meanAgain look at the graph

Group WorkCheck Chebyshevs theorem. What are the similarities and differences between them?Potential AnswerEmpirical rule gives a stronger statement than Chebyshevs theorem in it gives definite percentages, not just lower limits.Group Work: Creating the normal distribution curveThe yearly corn per acre on a particular farm is normally distributed with mean = 60 and standard deviation of 7. Create the normal distribution curve.Answer 49 56 63 60 67 74 81Group Work: Empirical RuleThe playing life of a Sunshine radio is normally distributed with mean 600 hours and standard deviation of 100 hours. What is the probability that a radio selected at random will last from 600 to 700 hours?34.1%TechnologyWhat are some applications of normal distribution curve?It gives us how reliable are the dataUsed for control chartsControl ChartsIt is useful when we are examining data over a period of equally spaced time interval or in some sequential order.Example of Control Chart

How to Make a control chart of the random variable xExample:Day123456789101112131415x11202523161982517202329181410Answer

Interpreting control chartMs. Tamara of the Antlers Lodge examines the control chart for housekeeping. During the staff meeting, she makes recommendations about improving service or, if all is going well, she gives her staff a well-deserved pat in the back. Determine if the housekeeping process is out of control

AnswerHomework PracticePg 244 #1-7, 11,13Standard Units and Areas Under the Standard Normal DistributionNote:ScenarioSuppose you have two students comparing their test result in two different classes.

Jack got a 76 while the class average was a 66

Jill got an 82 while the class scored 72

Both scored 10 points above the mean.

How do we determine who did better in respect to their peers in the class?

The AnswerYou have to use the z test or z score because you want to know the position in term of the standard deviation away from the mean.What is z test or z score?Note:Unless otherwise stated, in the remainder of the book we will take the word average to be either the sample arithmetic mean or population mean.Calculating z scoresSuppose the company states that their average large pizza is 10 inches in diameter and the standard deviation is 0.7 inches. A customer ordered a pizza and found out it is of 7 inches. Assume the pizza follows a normal distribution. If the size of the pizza is below three standard deviation, the company would be in danger of getting customer complaints and have a bad name in its company.

How many standard deviation s from the mean is 7? Is the company going to be in trouble?AnswerGroup Work(Fake Data) National weigh average for men at 58 is 139 lbs with a standard deviation of 15.7. If you are 58 and weigh 171 lbs. Find the z score. Should you be concerned?AnswerQuestion:How would you find the raw score x if you are given the z score, mean and standard deviation?AnswerGroup WorkBased on your experience, it takes you on average of 15 minutes to walk to school with standard deviation of 2 minutes.

A)On one particular day, it took you 12 minutes to get to school. What is the z score? Is the z value positive or negative? Why?

B) What would be the commuting time corresponding to a standard score of z=-2.5?AnswerA) z=-1.5. It is negative because it is shorter than the expected time.

B) 10 minutesStandard normal distributionArea under the curve. The left tail ruleLook at Appendix II Table 5 on A22Group ActivityWhat is the value for and sketch the graph:A) z=1.18

B) z=-2.35

C) z=0.56

D) z=-3.34Group Activity: challengeWhats the value for:

A) z>1.25

B) values in between z=1.21 and z=2.36

C) values to the right of 0.95

D) values in between z=-2.75 and z=1.38Homework practicePg 256 #1-48 eoeAreas under any normal curveHow do you find the z score/z value?AnswerExample:AnswerGroup WorkA typical iphone have a battery life that is normally distributed with a mean of 18 hours with a standard deviation of 2 hours. What is the probability that an iphone will have a battery life that is greater than 24 hours?AnswerTech notes finding area under the curve in between z valuesFind x given probabilityA company sells a hybrid battery that has a mean life of 8 years with standard deviation of 2 years. How long can the company guarantee its warranty such that they do not want to refund on more than 5% of the hybrid battery?AnswerGroup WorkA company sells a product that has a mean shell life of 60 month with the standard deviation of 5 months. How long can the company guarantee its warranty such that they do not want to refund on more than 7% of the product?Answeryou have to use z=-1.48

-1.48*5+60=52.6 monthsExample: Find zFind the z value such that 90% of the area under the standard normal curve lies between z and zAnswerGroup workFind the z value such that 80% of the area under the standard normal curve lies between z and zGroup workFind the z value such that 75% of the area under the standard normal curve lies between z and z

Tech note for finding x valueCheck for normalityCheck for normality502317607573252730465313614787783781762936486255Homework PracticePg 267 #1-30 eoe, 31Normal Approximation to the Binomial DistributionNormal approximation to the binomial distributionExample:AnswerImportant note about binomial approximationRemember how to graph the binomial distribution? Remember how you have to incorporate the 0.5?For example when r=5, you draw the binomial distribution from 4.5 to 5.5.How to make the Continuity CorrectionNormal ApproximationThe owner of a new apartment building must install 25 water heaters. From past experience in other apartment buildings, she knows that Quick Hot is a good brand. A Quick hot heater is guaranteed for 5 years only, but from the owners past experience, she knows that the probability it will last 10 years is 0.25.

A) What is the probability that 8 or more of the 25 water heaters will last at least 10 years?B) How does this result compare with the result we can obtain by using the formula for the binomial probability distribution with n=25 and p=0.25?C) How do the results of part a and b compare?AnswerGroup Work:The Denver Post stated that 80% of all new products introduced in grocery stores fail (are taken off the market) within 2 years. If a grocery store chain introduces 66 new products, what is the probability that within 2 yearsA) 47 or more fail?B) 58 of fewer fail?C) 15 or more succeed?D) fewer than 10 succeed?AnswerGroup WorkIt is estimated that 3.5% of the general population will live past their 90th birthday. In a graduating class of 753 high school seniors, what is the probability thatA) 15 or more will live beyond their 90th birthday?B) 30 or more will live beyond their 90th birthday?C) Between 25 and 35 will live beyond their 90th birthday?D) more than 40 will live beyond their 90th birthday?AnswerUse the examples and my answers to find out the pattern of the wording!Homework PracticePg 278 #1-14 eoo