1. Lecture 13 Sections 5.4 – 5.6 Objectives:
Random Variables and Sampling Distributions
Discrete random variables
Continuous random variables
Sampling Distributions
The Central Limit Theorem
Sampling distribution of the sample mean
Sampling distribution of the sample proportion

2. Random Variables
Note that the outcomes of a random experiment can be numerical, but they do not have to be. Often we “mathematized” those latter outcomes by assigning numbers to them.
For instance, we could denote the outcome {H} by number 0 and the outcome {T} by number 1.
Measurements on outcomes associated with random experiments are called random variables.
Definition. Given a random experiment with a sample space S, a random variable is a real valued function defined over a sample space. That is, x: S → R.
Example Consider the experiment of tossing two coins. Then, S={HH,HT,TH,TT}. Let x= # of heads observed. Then, x(HH)=2, x(HT)=1, x(TH)=1, x(TT)=0.

3. Types of Random Variables
There are two types of random variables:
1) Discrete:
A random variable is discrete if the number of possible values it can take is finite (e.g., {0,1}) or countably infinite (e.g., all nonnegative integers {0,1,2,…}).
Ex. # of heads, # of flips to observe a head, # of children, etc.
2) Continuous:
A random variable is continuous if it can assume any value from one or more intervals of real numbers. The possible values of a continuous random variable are uncountably infinite.
Ex. age, weight, humidity, temperature, etc.

4. Probability Distributions
The mechanism for assigning probabilities to events defined by random variables is to use either a mass function (for discrete random variables) or a density function (for continuous variables).
Example. n=20 items are randomly selected from a large lot. Let
A={at most one of the sampled items fail}
B={none of the sampled items fail}
C={exactly one item fail}
D={at least one item fail}.
Let x=# of items that fail.
Then,
A → x≤1
B → x=0
C → x=1
D → x≥1
If x~ B(n=20, π =0.05), then
P(A)=
P(D)=

5. Mean and Variance of Random Variables
1) The mean of a variable is
if x is a discrete variable
if x is a continuous variable.
2) The variance of a variable is

6. Sampling Distributions
We have seen the good statistical methods in Chapter 4 to take a representative sample. We took a sample to gain information about population. We use statistics to make inference about population parameters.

7. Sampling Distributions
There are many different samples that you can take from the population. Statistics can be computed on each sample. Since different members of the population are in each sample, the value of a statistic varies from sample to sample. This is called sampling variability.
The sampling distribution of a statistic is a distribution of values taken by the statistic in all possible samples of the same size from the same population. We can then examine the shape, center and spread of the sampling distribution.

8. Properties of Sampling Distributions
The sampling distribution of a statistic often tends to be centered at the value of the population parameter estimated by the statistic.
ii) The spread of the sampling distributions of many statistics tends to grow smaller as the sample size n increases.
iii) As the sample size n increases, sampling distributions of many statistics become more and more bell-shaped (more and more like normal distributions).

9. Sampling Distribution of the Sample Mean
We take many random samples of a given size n from a population with mean m and standard deviation s.
Some sample means will be above the population mean m and some will be below, making up the sampling distribution.
Sampling distribution of “x bar”
Histogram of some sample averages

10. Sampling distribution of x bar
For any population with mean m and standard deviation s:
The mean, or center of the sampling distribution of , is equal to the population mean m : mx = m.
The standard deviation of the sampling distribution is s/√n, where n is the sample size : sx = s/√n.
Sampling distribution of x bar
s/√n m

11. Note: When a population distribution is normal , the sampling distribution of is also normal, regardless of the size of the sample.
Example: A brand of water-softener salt comes in packages marked "net weight 40lb". The company that packages the salt claims that the bags contain an average of 40 lb of salt and that the standard deviation of the weights is 1.5 lb. Furthermore, it is known that the weights are normally distributed.
Find the probability that the weight of one randomly selected bag of water softener salt will be 39 lb or less, if the company's claim is true.
b. Find the probability that the mean weight of 10 randomly selected bags of water softener salt will be 39 lb or less, if the company's claim is true.

12. The Central Limit Theorem
The sampling distribution of can be approximated by a normal distribution when the sample size n is sufficiently large, irrespective of the shape of the population distribution. The larger the value of n, the better the approximation.
Population with strongly skewed distribution
Sampling distribution of for n = 2 observations
Sampling distribution of
for n = 25
observations
Sampling distribution of for n = 10 observations

13. Example
Achievement test scores of all high school seniors in a certain state have mean 60 and variance 64. A random sample of n=100 students from one large high school had a mean score of 58.
Is there evidence to suggest that this high school is inferior? Calculate the probability that the sample mean is at most 58.
b. What is the probability that the score test on the achievement test of a randomly chosen student is at most 58?

14. The Sampling Distribution of the Sample Proportion
Suppose that we wish to use samples of size n to estimate the probability of success (π ), e.g., π = the proportion of defectives being made by a certain process. Then, the estimate of π is given by
where x = # of successes (i.e., # of 1s).
Note that p is called the sample proportion.

15. Sampling Distribution of p
The mean and standard deviation of the sample proportion p are given by
and
For a large enough n, the sampling distribution of p is approximately Normal. In general, the normal approximation is best when both nπ ≥ 5 and n(1−π ) ≥ 5.
Note 1: The sample proportion can be thought of as a mean of n independent dichotomous outcomes. So, the CLT implies that the sampling distribution of p is approximately normal for a large enough n.
Note 2: Some books use 10 or 15 instead of 5 for a guide of “large enough” n.

16. Example
A certain process constantly generates an average about 5% nonconforming products. Samples of size 100 are taken each day to test whether the 5% rate has changed.
What is the probability that the estimated proportion is less than 6%?