Stat - Exam #2 Flashcards
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?
- Confidence Interval Estimation = give an estimate of the value of the UNKNOWN population parameter;
- Hypothesis Testing = Claim about a population, then sample data are collected and use to test this claim
Which standard deviation is ALWAYS known?
- Sample!;
- 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
What are the Estimators for population parameters?
Location = Mean, Median, Mode
Spread = Standard Deviation, Range, IQR
What are the properties of a good estimator?
- Unbiased = expected value of estimator equals value of parameter;
- Consistent = larger sample makes estimator more accurate;
- Efficient = estimator has the smallest standard deviation
What is a Confidence Interval Estimate?
Range of values if an interval on the real number line;
-Expresses the natural uncertainty in the estimate
What is a Confidence Level?
The proportion of confidence intervals calculated from a large number of random samples that contain the value of the population parameter
Denoted = CL;
Common Values = 90%, 95%, 99%;
Decided by the researcher
What is the Significance Level?
The area outside of the region of confidence;
Denoted = sigma Calculated = {1 - (CL/100)}
What is a Critical Value?
The pair of values that bound the region of confidence
Denoted = (+/- z_alpha/2’), (+/- t_alpha/2, n-1)