Week 11- Stat
Central limit theorem
If X had any distribution not normal then the shape of the sampling distribution of X bar is approximately normal: as long as n is greater than 30 ONLY ABOUT SHAPE RESULTS
The symbol we use to represent the mean of the random variable X-bar
Mu with subscript X-bar
What effects the standard error
N → as n increases sigma x bar decrease As sigma x increases (population standard deviation) your sigma x bar increases
Statistic
Number that describes the sample (sample mean, x bar), average lifetime
Parameter
Number that summarizes the population (population mean)
The set of all possible sample means from all possible samples of size n from the population is known as the
Sampling distribution of x bar
Sample
Subset of population that you select
Sampling distribution case 2
The shape of the distribution is not normal The shape of the sampling distribution of X is approximately normal if n is large enough
Sampling distribution case 1
The shape of the distribution of X is normal The shape of the sampling distribution is also normal Means are closer together
Relationship between parameters and statistics
We use statistics to estimate or test parameters
The confidence interval is affected by
outliers
What is in every confidence interval we make
sample mean
As n increases, the mean of the random variable X-bar
stays the same
The Central Limit theorem tells us important results that pertain to
the shape (type) of the distribution of x bar
To be 90% confident add and subtract
1.645 standard errors
If your confidence interval is 95%, what is the value of Z that goes into the confidence interval formula
1.96
To be 95% confident, add/subtract
1.96 standard errors ('about 2')
To be 99% confident, add/subtract
2.58 standard errors
Factors affecting MOE
Confidence level (More confident, Confidence level increases, Z increases, MOE increases) Sample size (n) (n increases, MOE decreases, More data, more precision) Population standard deviation (As sigma x increases, MOE increases)
Sampling distribution
Find the distribution of all possible values of the sample statistic (from all possible samples of size n)
If X does not have a normal distribution, the shape of the random variable X-bar is
approximately normal if n > 30
CLT is an
approximation
As x-bar increases, the standard error of the random variable
decreases
If X has a normal distribution, the shape of the random variable X-bar is
exactly normal for any n
The margin of error of a confidence interval gets larger if the confidence level _____________ (assume all else stays the same
increases
The margin of error is ___________ if the standard deviation of the population increases (assume all else stays the same.)
larger
A confidence interval for the mean is known as a range of ________ values for the population mean
likely
How large does n generally have to be in order for the Central Limit Theorem to take effect? (Assume X does not have a normal distribution.)
n>30