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What is the Sampling Distribution of the Statistics?

-Stats calculated from a sample varies with each sample;
-Variation because each stat is a random variable that follows some probability density CURVE with a LOCATION and SPREAD;


What is Sampling Distribution?

A probability density curve of all possible values of a statistic computed for a sample size (n).;
-Focus on the Population Means


What is the Law of Large Numbers?

As the sample size gets LARGER, the difference between the sample average and the population mean gets SMALLER


What is affected by SAMPLE SIZE with normally distributed data?

-Normally distributed, the MEAN of sample average is NOT affected by sample size, but the standard deviation of the sample average IS affect by size


What is Sampling Distribution of Sample Average?

If data are distributed normally with mean (u) and standard deviation (sigma), then the average of a sample of size (n) with be distributed normally with mean (u) and standard deviation [sigma/(sq. rt of n)]


IF/THEN of Sampling Distribution of Sample Average

IF: x has shape NOR with location (u) and spread (sigma)
THEN: x-bar has shape NOR with location (u) and spread {sigma/(sq. rt of n)}


What is the Standard Error of the Mean?

The standard deviation of the same average;
— sigma_x-bar = sigma/(sq. rt of n)


How do you find the shape of a sample for data NOT normally distributed?

-The sample average and the sample standard deviation can be calculated, but the shape is determined from a z-curve and the z-table = Central Limit Theorem


What is the Central Limit Theorem?

- When there are at least 30 data points (any shape, mean u, and standard deviation) the SAMPLE AVERAGE will...
1. follow the NORMAL shape
2. have mean (u) — same as population;
3. and have standard deviation {sigma/(sq. rt. of n)}


LESS than 30 data points

-Unknown shape;
-Mean (u);
-Standard deviation {sigma/(sq. rt. of n)}


Greater than or Equal to 30 data points

-NORMAL shape;
-Mean (u);
-Standard deviation {sigma/(sq. rt. of n)}


Difference in the Sampling Distribution and Central Limit Theorem

Sampling distribution of the mean deals with the location and spread of the sample average;
-The central limit theorem deals only with the SHAPE of the sample average


What is the population standard deviation is UNKNOWN?

-Use the sample standard deviation to calculate confidence interval estimates of a population parameter;
-The sample standard deviation can be calculated ANYTIME there is a sample


What is used to get degrees of freedom and critical values of the sample test?

The t-table


What do you calculate when the population standard deviation is UNKNOWN?

-Replace the populations standard deviation with with SAMPLE standard deviation = t-transforation that yields a t-statistic


What is a t-transformation?

Converts a sample average into a t-statistic

t = (x-bar - u) / [s/(sq. rt of n)]


What is t-distribution?

If a simple random sample of size (n) is taken from a population that follows the normal distribution, then the t-statistic follows the t-distribution with (n-1) degrees of freedom


What is a t-statistic?

t _ (sigma/2), (n-1) =

- sigma/2 = gives the area in one tail = Column of t-table;
- n-1 give the degrees of freedom = Row of t-table


How do you use the t-table?

-Need to know the area under the tail and the degrees of freedom;
-If the exact are not in the table, follow the practice of always going down to the next lower degrees of freedom in the table;
-NOTE: the LAST row of that -table is the same as the z-table


What is a reasonable value for the population mean?

A CONFIDENCE INTERVAL gives a set of values that are reasonable choices for the population mean based on the information in the SAMPLE data


Where does the level of confidence come from?

The NORMAL probability curve


What is Inferential Stats?

Use the information from a sample to make conclusions about the population


What is an Interval Estimate?

-Value of sample stat is very seldom the exact population parameter, but pretty close;
-Calculate a sample stat and an INTERVAL indicating how close the stat is to the population parameter;
**Central to Inferential Stats


What are the major methods of Inferential Stats?

1. Confidence Interval Estimation = give an estimate of the value of the UNKNOWN population parameter;

2. Hypothesis Testing = Claim about a population, then sample data are collected and use to test this claim


Which standard deviation is ALWAYS known?

- Population is not usually known in everyday practice


What is required when the population standard deviation is UNKNOWN?

Requires the used of a t-value
(Z-scores are only applicable when pop. standard deviation is already known)


What is the Point Estimate of a population parameter?

The value of the sample statistic used to estimate the population parameter


What is the Point Estimate of of the population MEAN?

The value of the sample average;
-BEST estimator of the population mean

Sample Average = POINT ESTIMATOR
Actual value of Sample Average = POINT ESTIMATE


What is the Point Estimate of the population STANDARD DEVIATION?

The value of the sample standard deviation


What values are estimated?

ONLY the values of the population parameters, NEVER the values of the sample statistics