Download 6.1 Central Limit Theorem Notes

Name: ___________________________
Teacher: _______________
Central Limit Theorem
Parameter: number that describes the ___________________________. It is a _____________
___________________ but we do not know its value because __________________________
____________________________________________________________________________.
Statistic: number the describes a ________________________. It is known when we have
_____________________________________________________________ but it can change
___________________________________________________________________________.
REMEMBER
Statistics come from Samples
Parameters come from Population
NOTE
− Mean of a Population:
______
− Mean of a Sample:
______
More Definitions
Sample means: ________________________________________________________________
_____________________________________________________________________________.
Sampling Distribution: _________________________________________________________
_____________________________________________________________________________.
− The variability of a statistic is described by ________________________________________
_____________________________________________________________________________.
− If you increase your ___________________________________________________________
the variability in the distribution of ___________________________________________.
− We are about to examine the distribution of sample means.
− This is also referred to as the ____________________________________________________.
Name: ___________________________
Teacher: _______________
Let’s experiment . . .
− Bowling scores are normal distributed with a mean of 160 and standard deviation of 20.
−
Is it normal to bowl a 139? ______
−
Is it normal to bowl a 225? ______
− Let’s have the calculator simulate some bowling scores
− CALCULATOR COMMAND: MATH ⇒ PRB ⇒ #6 ⇒ randNorm(______,______)
− We want to be able to look at the average score from 5 bowling games.
− First let’s have the calculator simulate 5 games with just one command:
− CALCULATOR COMMAND: MATH ⇒ PRB ⇒ #6 ⇒ randNorm(_____,_____,___)
− Now we want to find the average score from bowling 5 random games.
− CALCULATOR COMMAND: 2ND ⇒ STAT ⇒ MATH ⇒ #3
mean(
− MATH ⇒ PRB ⇒ #6
mean(randNorm(_______,_______,_____))
− Round your average score of 5 games to the nearest whole number. _____________
Looking at all of our sample means in L1, what do you notice?
− All of the scores were not exactly _______, however the numbers ______________________
_______________________________ when we were just calculating random scores of 1 bowling game.
● The mean of the sample means should be equal to the ____________________________.
● The mean of the population and mean of our sample mean is ______________________.
● If we used even bigger samples with more people the mean would be _______________.
What about the standard deviation?
− __________________________________________________________________________.
Name: ___________________________
Teacher: _______________
Reflect . . .
In our bowling example:
− What was the sample size for each sample taken? ________________
− What were the sample means? __________________________________________________
− Explain how the variability of the sample distribution was less than the variability of the population.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
The Central Limit Theorem
Given:
1. A random variable _____ has a distribution with a mean _____ and a standard deviation ____.
2. Samples of size _____ are selected from this population and the ________________________ are
calculated.
Conclusions
The Central Limit Theorem states that if we consider all ________________________________
____________________________ from a population with mean _____ and standard deviation _____
and we look at the sample means from these samples:
1. The mean of the sample mean, ______, is the ______________________________________.
2. The standard deviation of the sample mean is:
3. As the sample size ___________________________, the distribution of the sample mean will
___________________________________________________________________.
Name: ___________________________
Teacher: _______________
Example 1: Rat weights are normally distributed with μ = 8 lbs. and σ = 3 lbs.
− Identify Sample Mean:
− Calculate Sample Standard Deviation
− Run normalcdf
1) I randomly select 1 rat. What is the probability that it weighs less than 6 lbs.? _____________
2) I randomly select 30 rats. Find the mean and SD of the sampling distribution of 30 rats.
_____________________________________________________________________________
3) What is the probability that the 30 rats have an average weight less than 6 lbs.? ____________
Example 2: ACT scores are normally distributed with µ = 18.6 and σ = 5.9.
1) Find the probability that a randomly selected student has a score higher than 21. ___________
2) Find the probability that 10 randomly selected students have an average score greater than 21. ______
3) If 20 students are randomly selected, what is the probability that the average score is greater than 21?
___________
Example 3: Women's heights are normally distributed with mean = 65 in. and std dev = 2.5 in.
1) Find the probability that a randomly selected women is between 63 and 65 inches.___________
2) Find the probability that 10 randomly selected women have an average height between 63 and 65
inches. ___________
3) Find the probability that 25 randomly selected women have an average height between 63 and 65
inches.____________
Name: ___________________________
Teacher: _______________
What is original population isn’t normally distributed?
______________________________________________________________________________
______________________________________________________________________________
____________________________________________________________________________