7 Chapter Sampling & Sampling Distributions
Sampling Distributions Basics
Inferential Statistics
The area of statistics in which conclusions or statements are made about a population of interest Inferences are based on the sample data obtained from the population
Sampling Distributions Basics
Parameter
Numerical descriptive measurement of a population Asumed to be known In actuality, parameters are never (or rarely ever) known Take a sample to make inferences about the population
Sampling Distributions Basics
The sample results provide estimates of the values of the population characteristics With proper sampling methods, the sample results can provide "good" estimates of the population characteristics
Simple Random Sample
A simple random sample of size n from a finite population of size N is...
Simple Random Sampling (cont)
Sampling with replacement
Sampling without replacement is used more often. Why? In large sampling projects, computer generated random numbers are used to automate the sampling process
Sampling Distributions Basics
Sample Statistics
Numerical descriptive measures obtained from the sample Examples:
We will examine the properties of these statistics b/c these statistics have distributions themselves
Sampling Distribution of the Mean
Normal Distribution
Function of two parameters:
Want to estimate
Use mean, median, mode, etc. Which one would be closer to the "true" value?
Sampling Distribution of the Mean Example
Suppose we KNOW the TRUE average height of students in the class is 68 inches. If we number each student from 1-35 and randomly draw samples and measure their height
Sampling Distribution of the Mean (continued)
That is:
As the sample size increases, the mean of the means will approach , the population mean As the sample size increases, the spread (variation) will decrease As the sample size increases, the shape of the distribution of the means will become more bell-shaped (Normal)
Sampling Distribution (continued)
FACT
If all possible samples of size n are taken from the population, and the test statistic is calculated for each sample, then a sampling distribution can be determined along with its center, dispersion, and shape.
Sampling Distribution (continued)
Sample statistics
Properties of Point Estimates
Unbiased
Properties of Point Estimates Unbiasedness
Examples
E( x ) E (s )
2 2
Properties of Point Estimates (continued)
The standard deviation of an estimator is called the standard error of the statistic or estimate. The best estimate is one that has minimum variance and is unbiased.
Sampling Distribution of the Mean
In repeated sampling, x will be approximat ely normal with
2
a mean
n (population mean) and 2 is the true variance (population variance ).
X
and variance
2 X
, where is the true mean
Notationally, the sampling distribution for the mean is x~N where
x, 2
n
n is often called the standard error of the mean.
Sampling Distribution of the Mean
If a random sample of n observations is taken from a population with a normal distribution, the sampling distribution of the mean will be a normal distribution.
Central Limit Theorem (CLT)
Central Limit Theorem (CLT) (continued)
Central Limit Theorem (CLT) (continued)
Central Limit Theorem (CLT) (continued)
Rule of Thumb:
If n 30, then the CLT applicable
is Given the sampling distribution of the sample mean, we can compute z-scores to find probabilities in the following manner:
z
x
x
x
x
x
/ n
Sampling Distribution of the Mean Example 1
A recycling plant compresses aluminum cans into bales. The weights of the resulting bales are known to have a mean of 100 pounds and a variance equal to 64 pounds squared. A simple random sample of 50 bales is taken. (a) What is the probability the bales will weigh on the average less than 104 pounds? (b) 89.2% of the heaviest bales will be expected to weight at least how many pounds?
Sampling Distribution of the Mean Example 1
Sampling Distribution of the Mean Example 1
Sampling Distribution of the Mean Example 2
The weights of cans of peaches from a production line have a mean of 16.8 ounces and a variance of 2.25 ounces. From each run of the process, 100 cans are selected randomly and weighed. What is the probability that the average weight of these cans is between 16.5 and 17.1 ounces?
Sampling Distribution of the Mean Example 2
Sampling Distribution of the Mean Example 3
The lifetime of a certain type of micro transistor is normally distributed about a mean of 156.83 hours; 1.5 percent of the transistors have a life time greater than 167.68 hours. (a) What is the standard deviation of the distribution of the life times of the transistors? (b) What is the probability that the average life time of a sample of 25 transistors is at least 155 hours?
Sampling Distribution of the Mean Example 3
Sampling Distribution of the Mean Example 3
Sampling Distribution of the Sample Proportion
Recall that p is the binomial parameter. If we have a binomial experiment with n identical trials and we want to estimate p, the proportion of "successes" in the n trials. For n trials, X = number of "successes" Point estimate for p is:
^ p
X n
Sampling Distribution of the Sample Proportion
If we take a large number of samples of size n from a population, ^ then if n is large, then the sampling distribution of p will be approximat ely normally distributed with
^ p
p and
^ p
p(1 p) n
^ p ~ N p,
pq n
Sampling Distribution of the Sample Proportion
What is "large enough"?
If np and n(1-p) are both greater than 5, then we say that n is large enough.
n
5 min(p,1 p)
Sampling Distribution of the Sample Proportion
Now we can standardize:
Sampling Distribution of the Sample Proportion Example 1 A series of tests conducted by Metric Systems, Inc. indicates that a particular type of switch manufactured by the company will operate 95% of the time correctly. If 110 switches are selected at random: (a) What is the probability the sample will have no more than 98% correctly working switches? (b) What is the value for the sample proportion of switches that will work correctly 65% or less of the time?
Sampling Distribution of the Sample Proportion Example 1
Sampling Distribution of the Sample Proportion Example 1
Sampling Distribution of the Sample Proportion Example 2
Rambling Motors produces only red and green cars. They sell the same number of each color, which implies that they sell 50% red cars and 50% green cars. Out of the next randomly selected 200 cars, what is the probably that fewer than 48% are red?
Sampling Distribution of the Sample Proportion Example 2
End of Chapter 7