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