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6-2: STANDARD NORMAL AND UNIFORM DISTRIBUTIONS. IMPORTANT CHANGE Last chapter, we dealt with discrete probability distributions. This chapter we will.

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Presentation on theme: "6-2: STANDARD NORMAL AND UNIFORM DISTRIBUTIONS. IMPORTANT CHANGE Last chapter, we dealt with discrete probability distributions. This chapter we will."— Presentation transcript:

1 6-2: STANDARD NORMAL AND UNIFORM DISTRIBUTIONS

2 IMPORTANT CHANGE Last chapter, we dealt with discrete probability distributions. This chapter we will deal with continuous distributions. We are not focused on the probability of a specific data value, instead we care about ranges.

3 Topics  Normal Distributions in General  Probability as an Area  Uniform Distributions  Standard Normal Distribution  Calculating Probability  Calculating Z-Score

4 Normality is based on standard deviation and mean. There is a formula that can be used to describe the curve based on these parameters, however, we will not need to use it in this course. MEAN Normal Distribution: A continuous probability distribution that is symmetric and bell- shaped.

5 Probability as an Area  The graph representing a continuous distribution is also known as a density curve.  The total area under the curve must equal 1  Every point has a height of 0 or greater  Using this information, we can use area to represent probability.  This will start to make sense within the context of problems.

6 Uniform Distribution  A distribution is uniform if its probability remains the same for the entire range of possibilities. P(x) x

7 Example Ace Heating and Air Conditioning Service finds that the amount of time a repairman needs to fix a furnace is uniformly distributed between 1 and 5 hours. Find the probability that it takes more that 3.5 hours. 5.25 time Prob 0 1.375

8 You Try! A power company provides electricity with voltage levels that are uniformly distributed across 123.0- 125.0. Find the probability that a randomly selected voltage is greater than 124.5. 125.0.5 voltage Prob 0.25 123.0

9 Suggested Practice from p.261+  Uniform Distribution: 5-8  Find Probability from Left: 9, 17, 19  Find Probability from Right: 10, 21, 23  Find Probability in the Middle: 12, 25, 29  Find z Score from Left: 13, 50  Find z Score in Middle: 51  Find z Score from Right: 42, 43

10 Standard Normal Distribution  The standard normal distribution is a special case of the normal distribution in which the mean is 0 and the standard deviation is 1. 0123-2-3 z Scores AREA

11 Area and z Scores  z Score: As it was before z-scores represent distance on the horizontal scale (# of standard deviations from mean).  Area: The region under the curve bounded by a specific parameter or parameters.

12 Calculating Probability  Just like the uniform distribution, the area under the curve represents probability.  Calculating area is much more difficult with a curve, so we will refer to table A-2 which does the calculations for us.  The table refers to the area under the curve up to the specific z Score from the LEFT  DRAW A PICTURE FOR EVERY PROBLEM!!!

13 Example  A company that makes thermometers realizes that their product is not completely accurate. When the temperature is actually 0°, it sometimes reads slightly above or slightly below 0°. They find that this range is normally distributed with a mean of 0° and standard deviation of 1°. Find the probability that the thermometer reads less than 1.27°. 0.8980

14 You try!  A new card game called 3’s has a normal distribution for earnings, with the mean winnings being $0, and a standard deviation of a $1. What is the probability of losing more than $1.50? 0.0668

15 Question  What if they ask you to find area from the right?  Since the area is equal to 1, you can find the probability from the left (B) and the area from the right is A = 1 – B

16 You try!  Using the previous thermometer example, find the probability of randomly selecting one thermometer that reads above -1.23°. 0.8907

17 Example  Using the thermometer example, find the probability that the temperature is between -2.00° and 1.50°. 0.9104

18 Big Note!  If the area is bound between two numbers, find the probability from the left for both values, and subtract!  Remember area, like probability cannot be negative!

19 Suggested Practice from p.261+  Uniform Distribution: 5-7  Find Probability from Left: 9, 17  Find Probability from Right: 10, 21  Find Probability in the Middle: 12, 25  Find z Score from Left: 13, 50  Find z Score in Middle: 51  Find z Score from Right: 42, 43

20 Homework Quiz 0.7745

21 Finding the z Score, Given Probability  From the left:  Find the given probability in the table and figure out which z Score corresponds with it.  Bounded on both sides:  Treat each end separate  From the right:  Find the z Score that goes with the complement

22 Example  Using the thermometer example from earlier in the section, find the temperature that would represent the 89 th percentile (the temperature separating the bottom 89% from the top 11%). 1.23

23 Big Note!  If the area you are looking for in the table cannot be found exactly, but you see 2 z Scores that produce areas slightly above and slightly below that value, then just take the z score closest to the value you are looking for.  Example: Look for the z score that produces an area of.800.

24 Example  Use the “quarterback sneak” example to find the yardage that would represent the 99 th percentile. 2.33

25 Example  Using the thermometer example, find the z Scores that separate the bottom 5% and the top 5%. -1.645 and 1.645

26 One last thing…

27 Example -1.645

28 Suggested Practice from p.261+  Uniform Distribution: 5-7  Find Probability from Left: 9, 17, 19  Find Probability from Right: 10, 21, 23  Find Probability in the Middle: 12, 25, 29  Find z Score from Left: 13, 14, 50  Find z Score in Middle: 51  Find z Score from Right: 15, 16, 42, 43

29 Homework (graded for correctness)  Complete worksheet!


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