1. Continuous Distributions
Chapter 7
Continuous Distributions

2. Continuous random variables
Are numerical variables whose values fall within a range or interval
Are measurements
Can be described by density curves

3. Density curves Is always on or above the horizontal axis
Has an area exactly equal to one underneath it
Often describes an overall distribution
Describe what proportions of the observations fall within each range of values

4. Unusual density curves
Can be any shape
Are generic continuous distributions
Probabilities are calculated by finding the area under the curve

5. How do you find the area of a triangle?
P(X < 2) =

6. What is the area of a line segment?
P(X = 2) =
P(X < 2) =
.25

7. Is this different than discrete distributions?
In continuous distributions, P(X < 2) & P(X < 2) are the same answer.
Hmmmm…
Is this different than discrete distributions?

8. P(X > 3) = P(1 < X < 3) = .5(.375+.5)(1)=.4375
Shape is a trapezoid –
How long are the bases?
b1 = .5
b2 = .375
h = 1
P(X > 3) =
P(1 < X < 3) =
.5( )(1)=.4375
.5( )(2) =.5

9. P(X > 1) =
.75
.5(2)(.25) = .25
(2)(.25) = .5

10. P(0.5 < X < 1.5) =
.28125
.5( )(.5) =
(.5)(.25) = .125

11. Special Continuous Distributions

12. How do you find the area of a rectangle?
Uniform Distribution
Is a continuous distribution that is evenly (or uniformly) distributed
Has a density curve in the shape of a rectangle
Probabilities are calculated by finding the area under the curve
How do you find the area of a rectangle?
Where: a & b are the endpoints of the uniform distribution

13. What shape does a uniform distribution have?
The Citrus Sugar Company packs sugar in bags labeled 5 pounds. However, the packaging isn’t perfect and the actual weights are uniformly distributed with a mean of 4.98 pounds and a range of .12 pounds.
Construct the uniform distribution above.
What shape does a uniform distribution have?
What is the height of this rectangle?
How long is this rectangle?
1/.12
4.98
5.04
4.92

14. What is the length of the shaded region?
What is the probability that a randomly selected bag will weigh more than 4.97 pounds?
P(X > 4.97) =
.07(1/.12) = .5833
What is the length of the shaded region?
4.98
5.04
4.92
1/.12

15. What is the length of the shaded region?
Find the probability that a randomly selected bag weighs between 4.93 and 5.03 pounds.
What is the length of the shaded region?
P(4.93<X<5.03) =
.1(1/.12) = .8333
4.98
5.04
4.92
1/.12

16. What is the height of the rectangle? Where should the rectangle end?
The time it takes for students to drive to school is evenly distributed with a minimum of 5 minutes and a range of 35 minutes.
Draw the distribution
What is the height of the rectangle?
Where should the rectangle end?
1/35 5 40

17. b) What is the probability that it takes less than 20 minutes to drive to school?
P(X < 20) =
(15)(1/35) = .4286 5 40
1/35

18. c) What is the mean and standard deviation of this distribution?

19. How is this done mathematically?
Normal Distributions
Symmetrical bell-shaped (unimodal) density curve
Above the horizontal axis
N(m, s)
The transition points occur at m + s
Probability is calculated by finding the area under the curve
As s increases, the curve flattens & spreads out
As s decreases, the curve gets taller and thinner
How is this done mathematically?

20. A B 6 s s
Do these two normal curves have the same mean? If so, what is it?
Which normal curve has a standard deviation of 3?
Which normal curve has a standard deviation of 1?
YES B A

21. Empirical Rule
Approximately 68% of the observations fall within s of m
Approximately 95% of the observations fall within 2s of m
Approximately 99.7% of the observations fall within 3s of m

22. Suppose that the height of male students at BHS is normally distributed with a mean of 71 inches and standard deviation of 2.5 inches. What is the probability that the height of a randomly selected male student is more than 73.5 inches? 71
= .32
P(X > 73.5) = 0.16
68%

23. Standard Normal Density Curves
Always has m = 0 & s = 1
To standardize:
Must have this memorized!

24. State the probability statement Draw a picture Calculate the z-score
Strategies for finding probabilities or proportions in normal distributions
State the probability statement
Draw a picture
Calculate the z-score
Look up the probability (proportion) in the table

25. Write the probability statement Look up z-score in table
The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 hours. What proportion of these batteries can be expected to last less than 220 hours?
Write the probability statement
Draw & shade the curve
P(X < 220) =
.9082
Look up z-score in table
Calculate z-score

26. The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 hours. What proportion of these batteries can be expected to last more than 220 hours?
P(X>220) =
= .0918

27. The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 hours. How long must a battery last to be in the top 5%?
Look up in table to find z- score
P(X > ?) = .05
.95
.05
1.645

28. What is the z-score for the 63?
The heights of the female students at PWSH are normally distributed with a mean of 65 inches. What is the standard deviation of this distribution if 18.5% of the female students are shorter than 63 inches?
What is the z-score for the 63?
P(X < 63) = .185
-0.9 63

29. Will my calculator do any of this normal stuff?
Normalpdf – use for graphing ONLY
Normalcdf – will find probability of area from lower bound to upper bound
Invnorm (inverse normal) – will find z-score for probability

30. The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 hours. What proportion of these batteries can be expected to last less than 220 hours?
N(200,15)
P(X < 220) =
Normalcdf(-∞,220,200,15)=.9082

31. The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 hours. What proportion of these batteries can be expected to last more than 220 hours?
N(200,15)
P(X>220) =
Normalcdf(220,∞,200,15) = .0918

32. The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 hours. How long must a battery last to be in the top 5%?
P(X > ?) = .05
.95
.05
Invnorm(.95,200,15)=

33. The heights of female teachers at PWSH are normally distributed with mean of 65.5 inches and standard deviation of 2.25 inches. The heights of male teachers are normally distributed with mean of 70 inches and standard deviation of 2.5 inches.
Describe the distribution of differences of heights (male – female) teachers.
Normal distribution with m = 4.5 & s =

34. What is the probability that a randomly selected male teacher is shorter than a randomly selected female teacher?
4.5
P(X<0) =
Normalcdf(-∞,0,4.5, = .0901

35. Ways to Assess Normality
Use graphs (dotplots, boxplots, or histograms)
Normal probability (quantile) plot

36. Normal Probability (Quantile) plots
The observation (x) is plotted against known normal z-scores
If the points on the quantile plot lie close to a straight line, then the data is normally distributed
Deviations on the quantile plot indicate nonnormal data
Points far away from the plot indicate outliers
Vertical stacks of points (repeated observations of the same number) is called granularity

37. Why are these regions not the same width?
Consider a random sample with n = 5.
To find the appropriate z-scores for a sample of size 5, divide the standard normal curve into 5 equal-area regions.
Why are these regions not the same width?
Each region should have 20% of the area in them – the higher the curve the less width needed for 20%.

38. Why is the median not in the “middle” of each region?
These would be the z-scores (from the standard normal curve) that we would use to plot our data against.
Consider a random sample with n = 5.
Next – find the median z-score for each region.
Why is the median not in the “middle” of each region?
The median should divide the area into half. In this case, the first median would be at the 10th percentile, the second at the 30th percentile, etc.
-1.28
1.28
-.524
.524

39. What should happen if our data set is normally distributed?
Let’s construct a normal probability plot. The values of the normal scores depend on the sample size n. The normal scores when n = 10 are below:
Suppose we have the following observations of widths of contact windows in integrated circuit chips:
Sketch a scatterplot by pairing the smallest normal score with the smallest observation from the data set & so on
Normal Scores
Widths of Contact Windows
What should happen if our data set is normally distributed?

40. Notice that the boxplot is approximately symmetrical and that the normal probability plot is approximately linear.
Notice that the boxplot is approximately symmetrical except for the outlier and that the normal probability plot shows the outlier.
Notice that the boxplot is skewed left and that the normal probability plot shows this skewness.

41. Are these approximately normally distributed?
What is this called?
Both the histogram & boxplot are approximately symmetrical, so these data are approximately normal.
The normal probability plot is approximately linear, so these data are approximately normal.