Bell Ringer 1.Mrs. Hall’s class had a test score average of 81.2%. The standard deviation was 5.8%. Assuming the scores had a normal distribution, what range did approximately 68% of the students score? 2.Which of the following show approximately normal distribution? Choose all that apply. A. B.C.

Z Scores April 7, 2016

Z Scores  Also known as a standard score  Indicates how many standard deviations an element is from the mean  Can be placed on normal distribution curves (AKA bell curves)  Range from -3 to +3  Used to compare results from a test to a normal population  A z-score can tell you where a person’s weight is compared to the average population’s mean weight  You need to know…  Mean, µ  Standard Deviation, ơ

Z Score Formula

SAT Scores

How to Read a Z-Table 1.Find the first digit/decimal place of your z-score on the left column. 2.Find the hundredths place for your answer for the z-score in the top row 3.Where they intersect gives you the decimal representation for what percent scored below you Therefore, on the SAT you scored above 63.68% of other test takers

Printer Prices  The price of printers in a store have a mean of $240 and a standard deviation of $50. The printer that you eventually chose costs $340.  What is the z-score for the price of your printer?  How many standard deviations above the mean was your printer? The price of my printer was 2 standard deviations above the mean price

Height  Joey is 63 inches tall. The mean height for boys at his school is 68.1 inches, and the standard deviation of the boys’ heights is 2.8 inches. What is the z-score for Joey’s height? Round to the nearest hundredth. What does the value mean? Using the z- table, what percent of boys at Joey’s school are shorter than Joey?  Z-Score:  The z-score means that Joey is 1.82 standard deviations below the average height for boys at his school.  Only 3.44% of the boys at Joey’s school are shorter than him.

Z-Scores and Probability  You can use the Z-table to determine the probability of an event happening.  Be careful with the question! It is different depending on if they are asking for less than or greater than!  Z-table is for values less than. If they ask for greater than, then you must subtract from z-table value from 1

Highway Speeds  Vehicle speeds at a highway location have a normal distribution with a mean of 65 mph and a standard deviation of 5 mph. What is the probability that a randomly selected car is going 73 mph or less?  Draw a sketch  Determine the z-score for 73 mph.  Z-score=1.60  Use a z-table to determine the probability.  There is a 94.52% chance the car is going less than 73 mph.

Pulse Rate  Suppose pulse rates of adult females have a normal curve distribution with a mean of 75 bpm and a standard deviation of 8. What is the probability that a randomly selected female has a pulse rate greater than 85?  Draw a sketch  Determine the z-score for 85 bpm.  Z-score=1.25  Use a z-table to determine the probability.  BE CAREFUL! We must subtract from 1!  The probability that a randomly selected female will have a pulse rate above 85 bpm is 10.56%.

Challenge: Swimming  A swimmer named Amy specializes in the 50-meter backstroke. In competition, her mean time for the event is 39.7 seconds, and the standard deviation of her times is 2.3 seconds. Assume that Amy’s times are approximately normally distributed. Estimate the probability that Amy’s time is between 37 and 44 seconds. Then estimate the probability that Amy’s time is more than 45 seconds. Probability that her time is between 37 and 44 seconds: Approximately 84.8% Probability that her time is more than 45 seconds: Approximately 3%