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##### Chapter 6 Normal Distributions

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**Characteristics of Normal Distribution**• The curve is bell-shaped, with the highest point over the mean • The curve is symmetrical about a vertical line through • The curve approaches the horizontal axis but never touches or crosses it • The inflection (transition) points between cupping upward and downward occur above**Let’s test your knowledge**Do these distributions have the same mean? Compare the two graphs, what can you tell me about the standard deviation?**Answer**• A) 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 rule**• For 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 mean • Approximately 95% of the data values will lie within two standard deviations on each side of the mean • Approximately 99.7% of the data values will lie within three standard deviations on each side of the mean**Group Work**• Check Chebyshev’s theorem. What are the similarities and differences between them?**Potential Answer**• Empirical rule gives a stronger statement than Chebyshev’s theorem in it gives definite percentages, not just lower limits.**Group Work: Creating the normal distribution curve**• The 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 81**Group Work: Empirical Rule**• The 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?**Technology**• If you have TI-83/TI-84 • Press the Y= key. Then, under DISTR, select 1:normalpdf and fill in desired values. Press the WINDOW key. Set Xmin to Finally, press the ZOOM key and select option 0: ZoomFit**What are some applications of normal distribution curve?**• It gives us how reliable are the data • Used for control charts**Control Charts**• It is useful when we are examining data over a period of equally spaced time interval or in some sequential order.**How to Make a control chart of the random variable x**• 1) Find the mean and standard deviation of the x distribution by • A) Using past data from a period during which the process was “in control” or • B) using specified “target” values for • 2) Create a graph in which the vertical axis represents x values and the horizontal axis represents time. • 3) Draw a horizontal line at height • 4) Plot the variable x on the graph in time sequence order. Use line segments to connect the points in time sequence order**Example:**Find the mean and the standard deviation (population) Find Find control limit**Answer**• Mean = 18.53 • Standard deviation = 5.82**Interpreting control chart**• Ms. 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**Answer**• Graph to the left: there are 9 consecutive days on one side of the mean. It is a warning signal. It would mean that the mean is slowly drifting upward from 19.3. • Graph to the right: We have data value beyond . We have two of three data values beyond . Both are warning signals.**Homework Practice**• Pg 244 #1-7, 11,13**Standard Units and Areas Under the Standard Normal**Distribution**Note:**• Normal distributions vary from one another in two ways: The mean may be located anywhere on the x axis, and the bell shape may be more or less spread according to the size of the standard deviation**Scenario**• Suppose 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 Answer**• You 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?**• Z value or z score gives the number of standard deviations between the original measurement x and the mean of the x distribution**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 scores**• Suppose 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?**Answer**• It is 4.28 below the mean, and therefore the company is in trouble.**Group Work**• (Fake Data) National weigh average for men at 5’8 is 139 lbs with a standard deviation of 15.7. If you are 5’8 and weigh 171 lbs. Find the z score. Should you be concerned?**Answer**• Answer may vary depending how you think of it. If you consider outlier is when z=2.5 or more then no, but if you say that he is not with the 95% (within 2 standard deviation) then he should. (based on empirical rule)**Question:**• How would you find the raw score x if you are given the z score, mean and standard deviation?**Group Work**• Based 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?**Answer**• A) z=-1.5. It is negative because it is shorter than the expected time. • B) 10 minutes**Standard normal distribution**• Standard normal distribution is a normal distribution with mean and standard deviation**Area under the curve. The left tail rule**• Look at Appendix II Table 5 on A22**Group Activity**• What is the value for and sketch the graph: • A) z=1.18 • B) z=-2.35 • C) z=0.56 • D) z=-3.34**Group Activity: challenge**• What’s 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=1.38**Homework practice**• Pg 256 #1-48 eoe**Example:**• Let x have a normal distribution with Find the probability that an x value selected at random is between (5 and 8).**Answer**• You have to convert the x into z score: • .6293-.2546=.3747**Group Work**• A 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?