1. 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. 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

5. 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

6. 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

7. 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

8. 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

9. 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

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

11. 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?

12. Normal distributions occur frequently.
Length of newborn child
Height
Weight
ACT or SAT scores
Intelligence
Number of typing errors
Chemical processes

13. 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

14. 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

15. Suppose that the height of male students at CHS 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%

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

17. 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

18. 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

19. 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

20. 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

21. What is the z-score for the 63?
The heights of the female students at CHS 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

22. The heights of female teachers at CHS 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 =

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

24. Will my calculator do any of this normal stuff?
Normalpdf – use for graphing ONLY
Y=normalpdf(x,µ,σ)
Normalcdf – will find probability of area from lower bound to upper bound (if starting from left, lower bound = -1x1099 )
Normalcdf(lower,upper,µ,σ)
Invnorm (inverse normal) – will find z-score from probability number
Z-score = Invnorm(probability)

25. Ways to Assess Normality
Use graphs (dotplots, boxplots, or histograms)
Use the Empirical Rule
Normal probability (quartile) plot on calculator

26. Normal Probability (Quartile) plots
The observation (x) is plotted against known normal z-scores
If the points on the quartile plot lie close to a straight line, then the data is normally distributed
Deviations from a straight line on the quartile 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

27. Are these approximately normally distributed?
Both the histogram & boxplot are approximately symmetrical, so these data are approximately normal.

28. Are these approximately normally distributed?
What is this called?
To create a Normal Probability plot, enter the data into L1 and then choose the 6th graph type in STATPLOT. L1 is x-axis and y-axis is theoretical z-scores for normal distribution
The normal probability plot is approximately linear, so these data are approximately normal.