1. Continuous Random Variables 2
Lecture 23
Section 7.5.4
Tue, Feb 26, 2008

2. Example
Now suppose we use the TI-83 to get three random numbers from 0 to 1, and then average them.
Let X3 = the average of the three random numbers.
What is the pdf of X3?

3. Example
The graph of the pdf of X3. 3 y
0.25
0.5
0.75 1

4. Example
The graph of the pdf of X3. 3
Area = 1 y
0.25
0.5
0.75 1

5. Example What is the probability that X3 is between 0.25 and 0.75? 3 y
0.25
0.5
0.75 1

6. Example What is the probability that X3 is between 0.25 and 0.75? 3 y
0.25
0.5
0.75 1

7. Example
The probability equals the area under the graph from 0.25 and 0.75. 3
Area = y
0.25
0.5
0.75 1

8. Hypothesis Testing (n = 3)
An experiment is designed to determine whether a random variable X has the distribution U(0, 1) or U(0.5, 1.5).
H0: X is U(0, 1).
H1: X is U(0.5, 1.5).
Three values of X3 are sampled (n = 3). Let X3 be the average.
If X3 is more than 0.75, then H0 will be rejected.

9. Hypothesis Testing (n = 3)
Distribution of X3 under H0:
Distribution of X3 under H1:
0.5
1.5 1
1.5
0.5 1

10. Hypothesis Testing (n = 3)
Distribution of X3 under H0:
Distribution of X3 under H1:
0.5
1.5 1
1.5
0.5 1

11. Hypothesis Testing (n = 3)
Distribution of X3 under H0:
Distribution of X3 under H1:
 =
0.5
1.5 1
 =
1.5
0.5 1

12. Example
Suppose we get 12 random numbers, uniformly distributed between 0 and 1, from the TI-83 and get their average.
Let X12 = average of 12 random numbers from 0 to 1.
What is the pdf of X12?

13. Example
It turns out that the pdf of X12 is nearly exactly normal with a mean of 1/2 and a standard deviation of 1/12.
N(1/2, 1/12) x
1/4
1/2
3/4

14. Verification Use Avg2.xls to generate 10000 pairs of values of X.
See whether about 75% of them have an average between 0.25 and 0.75.

15. Verification Use Avg3.xls to generate 10000 triples of values of X.
See whether about 85.94% of them have an average between 0.25 and 0.75.

16. Verification Use Avg12.xls to generate 10000 sets of 12 values of X.
See whether the data agree with the rule.

17. Preview of the Central Limit Theorem
We looked at the distribution of the average of 1, 2, 3, and 12 uniform random variables U(0, 1).
We saw that the shapes of their distributions was moving towards the shape of the normal distribution.

18. Preview of the Central Limit Theorem2 1 1

19. Preview of the Central Limit Theorem2 1 1

20. Preview of the Central Limit Theorem2 1 1

21. Preview of the Central Limit Theorem
Some observations:
Each distribution is centered at the same place, ½.
The distributions are being “drawn in” towards the center.
That means that their standard deviation is decreasing.
Can we quantify this?

22. Preview of the Central Limit Theorem2
= ½
2 = 1/12 1 1

23. Preview of the Central Limit Theorem2
= ½
2 = 1/24 1 1

24. Preview of the Central Limit Theorem2
= ½
2 = 1/36 1 1

25. Preview of the Central Limit Theorem2
Area = 0.20 1 1
n = 1

26. Preview of the Central Limit Theorem2
Area = 0.36 1 1
n = 2

27. Preview of the Central Limit Theorem2
Area = 0.432 1 1
n = 3

28. Preview of the Central Limit Theorem
When n = 12, we find
P(0.4 < X12 < 0.6) = normalcdf(.4,.6,.5,1/12) =

29. Preview of the Central Limit Theorem
This tells us that a sample mean based on three observations is more likely to be close to the population mean than is a sample mean based on only one or two observations.

30. Parameters and Statistics
THE PURPOSE OF A SAMPLE STATISTIC IS TO ESTIMATE A POPULATION PARAMETER.
A sample mean is used to estimate the population mean.
A sample proportion is used to estimate the population proportion.

31. Parameters and Statistics
Sample statistics are variable.
Population parameters are fixed.
In fact, a statistic is a random variable.
Therefore, it has a probability distribution.

32. Some Questions
We hope that the sample proportion is close to the population proportion.
How close can we expect it to be?
Would it be worth it to collect a larger sample?
If the sample were larger, we would expect the sample proportion to be closer to the population proportion.
How much closer? And how much larger?

33. Experiment
Survey finds drop in hand hygiene

34. Experiment
Person
Washes 1
Yes 11 No 21 31 41 2 12 22 32 42 3 13 23 33 43 4 14 24 34 44 5 15 25 35 45 6 16 26 36 46 7 17 27 37 47 8 18 28 38 48 9 19 29 39 49 10 20 30 40 50