chapter 18 – part ii sampling distribution for the sample mean
TRANSCRIPT
Chapter 18 – Part II
Sampling Distribution for the Sample Mean
Review
One Quantitative Variable– Age– Height
Review
Population parameters– ____________________________________
– ____________________________________
Review
Sample statistics– _______________________________________
– _______________________________________
Example
1994 Senators Population Parameter– _______________________________________
Sample of 10 Senators– _______________________________________
Example
Sample
1
2
3
4
5
SRS characteristics
Value of ___________________________
Changes from sample to sample. Different from population parameter.
– _______________________________
Long Term Behavior of Sample Mean
Repeat taking SRS of size n = 10.– ______________________________________
___________________________________
Sampling Distribution of Sample Mean
Summarize _________________________
__________________________________– Center (Mean)– Spread (Standard Deviation)– Shape
Sampling Distribution of Sample Mean
Mean (Center)
Sampling Distribution for Sample Mean
Standard Deviation (Spread)
Sampling Distribution for Sample Mean
Distribution of Population Values for Quantitative Variable is Normal
Three conditions1. Sample must be random sample
2. Sample must be independent values
3. Sample must be less than 10% of population.
Sampling Distribution for Sample Mean
When Population Values Come from a Normal Distribution
Sampling Distribution for Sample Mean
Distribution of Population Values for Quantitative Variables is non-normal.
Three conditions1. Sample must be random sample
2. Sample must be independent values
3. Sample must be less than 10% of population.
Sampling Distribution for Sample Mean
When Population Values Come from a Non-Normal Distribution.
Central Limit Theorem
As the sample size n increases, the mean of n independent values has a sampling distribution that tends toward a ______________________________.
Central Limit Theorem
Major Result in Statistics!
When Population Values Come from a non- Normal Distribution, Central Limit Theorem still applies when sample size is large.
How large does n need to be?
Depends on shape of population distribution– Symmetric: – Skewed:
68-95-99.7 Rule for Sampling Distribution of Sample Mean
In 68% of all samples, the sample mean will be between
68-95-99.7 Rule for Sampling Distribution of Sample Mean
In 95% of all samples, the sample mean will be between
68-95-99.7 Rule for Sampling Distribution of Sample Mean
In 99.7% of all samples, the sample mean will be between
Example #1
The height of women follows a normal distribution with mean 65.5 inches and standard deviation 2.5 inches.
If you take a sample of size 25 from a large population of women, use the 68-95-99.7 rule to describe the sample mean heights.
Example #1 (cont.)
Example #1 (cont.)
Example #1 (cont.)
Probabilities with the Sample Mean
We can also find probabilities with the sample mean statistic.
Example #1
Ithaca, New York, gets an average of 35.4 inches of rainfall per year with a standard deviation of 4.2 inches. Assume yearly rainfall follows a normal distribution.
Example #1 – cont.
What is the probability a single year will have more than 40 inches of rain?
Example #1 – cont.
Example #1 – cont.
What is the probability that over a four year period the mean rainfall will be less than 30 inches?
Example #1 – cont.
Example #2
Carbon monoxide emissions for a certain kind of car vary with mean 2.9 gm/mi and standard deviation 0.4 gm/mi. A company has 80 cars in its fleet. Estimate the probability that the mean emissions for the fleet is between 2.95 and 3.0 gm/mi.
Example #2 – cont.
Example #2 – cont.
Example #3
Grocery store receipts show that customer purchases are skewed to the right with a mean of $32 and a standard deviation of $20.
Example #3 – cont.
– Can you determine the probability the next customer will spend at least $40?
Example #3 – cont.
– Can you determine the probability the next 50 customers will spend an average of at least $40?
Example #3 – cont.