Warm Up – 5.20 - Tuesday The table to the left shows the amount of money made at six different dealerships in a single year. Give the five number summary B) Draw a boxplot to represent this data. C) Find the standard deviation (show work)

Z - scores A Z-score is a way of comparing two sets of data. Z-scores measure how many standard deviations a value is away from the mean. 𝑧= 𝑥− 𝑥 𝜎 A negative z-score means it is below the mean while positive is above the mean.

Example #1 You and your friend each got an 80% on a stats test. You are told the following: the two tests had a mean score of 75, in your class the standard deviation was 5 and the in your friend’s class standard deviation was 10. Who scored relatively better on the test, you or your friend?

Example #1 - Solution We can figure this out using z-scores. Your z-score Your friends z-score z= 80−75 5 z= 80−75 10 𝑧=1 𝑧= 1 2 You scored 1 standard deviation above the mean while your friend only scored half a standard deviation above the mean. You are doing relatively better.

Example #2 Stature (height) of adult males in the United States averages 69.0 in. with a SD of 2.8 in. How does Lebron James, who measures 80 inches tall, compare to the height of an average adult male living in the United States? Is Lebron’s height unusual?

Example #2 Lebron’s z-score: 𝑧= 80−69 2.8 =3.93 Lebron’s z-score is well above two which means Lebron is well above the mean height. He is unusually tall.

Converting z-scores back to raw scores IQ scores are known to have a mean of 100 and a standard deviation of 15. Candice’s IQ is 3.5 standard deviations above average. What is her IQ score and what can we interpret from this?

We set up the formula and solve to find Candice’s IQ 𝑧= 𝑥− 𝑥 𝜎 We know Candice’s z-score is 3.5. The mean is 100 and the standard deviation is 15. 3.5= 𝑥−100 15 Multiply by 15 52.5=𝑥−100 Add 100 152.5=𝑥 We can determine Candice has an unusually high IQ because she is more than 2 standard dev’s above the mean.

Example #3 A manufacturer of tires has a quality control policy that requires it to destroy any tires that are more than 1.5 standard deviations from the mean. The quality control engineer knows that the tires coming off the assembly line have a mean thickness of 10cm with a standard deviation of 0.7 cm. For what thickness will a tire be destroyed?

Example #3 - Solution The process is exactly the same as before. The z-score is 1.5, the mean is 10 and the std. dev is 0.7. ±1.5= 𝑥−10 0.7 Multiply by 0.7 ±1.05=𝑥−10 Add 10 𝑥=10±1.05 Tires more than 11.05 cm thick and less than 8.95 cm thick will be destroyed.

Z-scores!

Empirical Rule The Empirical Rule says, if I have a perfectly NORMAL CURVE (Bell shaped), 68% of the data will fall within one SD of the mean, 95% will fall within 2 SD, and 99.7% will fall within 3 SD of the mean.

Example #1 The mean weight of an adult american male is 180 pounds with a SD of 25 pounds. The weights are approximately normally distributed. Draw and label a normal curve

Empirical Rule 68% within ±1 SD. 95% within ±2 SD 99.7% within ±3 SD It will be important to know how much is left over in each tail section of a normal curve.

Example #2 The curve above shows the amount of money people raised for a fundraiser. What percent of people raised over $590 What percent of people raised between $290 and $590?

Example #2 – Solution A A) 50% of the data lies below 490. 34% lies between 490 and 590. Together this is 84%, leaving 16% of the people above $590.

Example #2 – Solution B B) 390 to 590 is ±1 𝑆𝐷. This is 68% of the people. The extra from 290 to 390 is 13.5%. In total we have 81.5%.