7.1

1. 7.1 Properties of the Normal Distribution

2. Between numbers • Sometimes we want to model a random variable that is equally likely between two limits • Examples • Choose a random time … the number of seconds past the minute is random number in the interval from 0 to 60 • Observe a tire rolling at a high rate of speed … choose a random time … the angle of the tire valve to the vertical is a random number in the interval from 0 to 360

3. Uniform Probability Distribution • When “every number” is equally likely in an interval, this is a uniformprobabilitydistribution • Any specific number has a zero probability of occurring • The mathematically correct way to phrase this is that any two intervals of equal length have the same probability

4. Example • For the seconds after the minute example • Every interval of length 3 has probability 3/60 • The chance that it will be between 14.4 and 17.3 seconds after the minute is 3/60 • The chance that it will be between 31.2 and 34.2 seconds after the minute is 3/60 • The chance that it will be between 47.9 and 50.9 seconds after the minute is 3/60

5. Example Continued • For the seconds after the minute example • Every individual value has probability 0 • The chance that it will be exactly 13.4 seconds after the minute (i.e. between 13.4 and 13.4) is 0/60 or 0 • The chance that it will be exactly 31.2 seconds after the minute (i.e. between 31.2 and 31.2) is 0/60 or 0 • The chance that it will be exactly 47.9 seconds after the minute (i.e. between 47.9 and 47.9) is 0/60 or 0 • We need to use a different way to specify the probabilities

6. Probability Density Function • A probabilitydensityfunction is an equation used to specify and compute probabilities of a continuous random variable • This equation must have two properties • The total area under the graph of the equation is equal to 1 (the total probability is 1) • The equation is always greater than or equal to zero (probabilities are always greater than or equal to zero)

7. Probability • This function method is used to represent the probabilities for a continuous random variable • For the probability of X between two numbers • Compute the area under the curve between the two numbers • That is the probability

8. The probability To 8 (here) From 4 (here) Probability • The probability of being between 4 and 8

9. Probability • An interpretation of the probability density function is • The random variable is more likely to be in those regions where the function is larger • The random variable is less likely to be in those regions where the function is smaller • The random variable is never in those regions where the function is zero

10. Less likely values More likely values Graph of Probability • A graph showing where the random variable has more likely and less likely values

11. Time Example • The time example … uniform between 0 and 60 • All values between 0 and 60 are equally likely, thus the equation must have the same value between 0 and 60

12. Time Example Continued • The time example … uniform between 0 and 60 • Values outside 0 and 60 are impossible, thus the equation must be zero outside 0 to 60

13. 1/60 Time Example Continued • The time example … uniform between 0 and 60 • Because the total area must be one, and the width of the rectangle is 60, the height must be 1/60

14. 1/60 Time Example Continued • The time example … uniform between 0 and 60 • The probability that the variable is between two numbers is the area under the curve between them

15. Normal Curve • The normalcurve has a very specific bell shaped distribution • The normal curve looks like

16. Normal • A normallydistributed random variable, or a variable with a normalprobabilitydistribution, is a random variable that has a relative frequency histogram in the shape of a normal curve • This curve is also called the normaldensitycurve (a particular probability density function)

17. Inflection Point • In drawing the normal curve, the mean μ and the standard deviation σ have specific roles • The mean μ is the center of the curve • The values (μ – σ) and (μ + σ) are the inflection points of the curve

18. Different Normal? • There are normal curves for each combination of μ and σ • The curves look different, but the same too • Different values of μ shift the curve left and right • Different values of σ shift the curve up and down

19. Normal • Two normal curves with different means (but the same standard deviation) • The curves are shifted left and right

20. Normal • Two normal curves with different standard deviations (but the same mean) • The curves are shifted up and down

21. “Normal” Properties • Properties of the normal density curve • The curve is symmetric about the mean • The mean = median = mode, and this is the highest point of the curve • The curve has inflection points at (μ – σ) and (μ + σ) • The total area under the curve is equal to 1 • The area under the curve to the left of the mean is equal to the area under the curve to the right of the mean

22. Normal Properties • Properties of the normal density curve • As x increases, the curve goes to zero … as x decreases, the curve goes to zero • In addition, the empirical rule holds • The area within 1 standard deviation of the mean is approximately 0.68 • The area within 2 standard deviations of the mean is approximately 0.95 • The area within 3 standard deviations of the mean is approximately 0.997

23. Normal Properties • The curve is symmetric about the mean • Because the curve is symmetric • The mean = median = mode = μ • The area under the curve to the left of the mean is equal to the area under the curve to the right of the mean

24. Normal Properties • The highest point of the curve is at x = μ • This can be seen in a previous chart • The total area is equal to 1 • This is a complex calculation • But it is true

25. Normal Properties • It has inflection points (where the concavity changes) at (μ – σ) and (μ + σ) • This can be seen in a previous chart • As x increases, the curve goes to zero … as x decreases, the curve goes to zero • This is clear from the chart also

26. Normal Properties • The empirical rule is true • Approximately 68% of the values lie between(μ – σ) and (μ + σ) • Approximately 95% of the values lie between(μ – 2σ) and (μ + 2σ) • Approximately 99.7% of the values lie between(μ – 3σ) and (μ + 3σ)

27. Reminder… • An illustration of the Empirical Rule

28. Normal Curve • The equation of the normal curve with mean μ and standard deviation σ is • This is a complicated formula, but we will never need to calculate it (thankfully)

29. Area under the curve • When we model a distribution with a normal probability distribution, we use the area under the normal curve to • Approximate the areas of the histogram being modeled • Approximate probabilities that are too detailed to be computed from just the histogram

30. Example • Assume that the distribution of giraffe weights has μ = 2200 pounds and σ = 200 pounds

31. Example • What is an interpretation of the area under the curve to the left of 2100?

32. Example • It is the proportion of giraffes that weigh 2100 pounds and less

33. How? • How do we calculate the areas under a normal curve? • If we need a table for every combination of μ and σ, this would rapidly become unmanageable • We would like to be able to compute these probabilities using just one table • The solution is to use the standard normal random variable

34. How? • The standard normal random variable is the specific normal random variable that has • μ = 0 and • σ = 1 • We can relate general normal random variables to the standard normal random variable using a Z-score calculation

35. Z- Score • If X is a general normal random variable with mean μ and standard deviation σ then is a standard normal random variable • This equation connects general normal random variables with the standard normal random variable • We only need a standard normal table

36. Example Continued • The area to the left of 2100 for a normal curve with mean 2200 and standard deviation 200

37. Z-Score • To compute the corresponding value of Z, we use the Z-score • Thus the value of X = 2100 corresponds to a value of Z = – 0.5 • In the next section, we will learn how to use this to find the probability!!!

38. Summary • Normal probability distributions can be used to model data that have bell shaped distributions • Normal probability distributions are specified by their means and standard deviations • Areas under the curve of general normal probability distributions can be related to areas under the curve of the standard normal probability distribution

39. The birth weights of full-term babies are normally distributed with mean 3,400 grams and standard deviation 505 grams • Draw a normal curve with the parameters labeled • Shade the region that represents the proportions of full-term babies who weigh more than 4,410 grams • Suppose that the area under the curve to the right of x = 4,410 is .0228. Provide two interpretations of this result.