Upgrade to remove ads
Only $35.99/year

6002-3

Terms in this set (17)

Definitions
Review
Random variable: Variable that has a single numerical value determined by chance for each outcome of an experiment
Does not have a single value, but instead has series of values, each with an associated probability
Probability distribution: A graph, table, or formula that gives the probability for each value of the random variable
New definition
Sampling distribution of the sample means: Probability distribution of the sample means that is obtained when we repeatedly draw samples of the same size n from the same population.
Example: Sampling Distribution
Consider population of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which are randomly selected with replacement.

Random variable: Suppose we randomly select a single digit and let x be the value of the selected digit; repeat this; X is a random variable (value depends on chance).
Probability distribution: Assuming the digits are randomly selected, the probability of each digit is P(x) = 1/10, which is a probability distribution.
Sampling distribution: Suppose that we now randomly select many different samples each of size n = 4; in each sample, calculate the mean x-, which is also a random variable (value depends on chance); probability distribution of the sample means x- is a sampling distribution.
Central Limit Theorem (CLT)
Given:
Random variable X has a distribution with mean μ and standard deviation σ.
Samples of all the same size n are randomly selected from the population of x values.
Conclusions:
As n increases, distribution of sample means x- will approach a normal distribution.
Mean of sample means will approach population mean μ.
Standard deviation of sample means will approach σn√; this is the standard error of the mean.
CLT: Common Practical Rules
For larger samples (n > 30), distribution of sample means can be approximated by a normal distribution.
If original population is normally distributed, then sample means will be normally distributed for any sample size n.
Review: Standard Error (SE)
Definition: The standard deviation of the sampling distribution (of all possible sample means) and not of the data itself
Notation: σx-=σ/√n
Property: As the sample size n increases, the standard error decreases.
Student t-Distribution
Developed by William Gosset, a Guiness Brewery employee who published under the pseudonym Student

Definition: Bell-shaped distribution typically associated with small sample sizes (n ≤ 30)
Formula depends on sample size (n):
t=x⎯⎯−μsn‾‾√
Conditions for use:
Small sample size n ≤ 30
Population standard deviation σ is unknown
Population has an essentially normal distribution
t-Distribution vs. Standard
Normal Distribution
Assume standard normal distribution where mean = 0
and standard deviation = 1.

Student t-Distribution Standard Normal (z) Distribution
Different for different sample sizes n Same regardless of sample size n
Wider symmetric bell shape that reflects greater variability expected with small samples Symmetric bell shape
Mean: t = 0 Mean: z = 0
Standard deviation: df‾‾√(df−2)
> 1 and varies with sample size n Standard deviation: σ = 1
Critical value: tdf,α2 Critical value: zα2
Degrees of Freedom (df)
df=n−1
Definition: Number of sample values that are free to vary after restrictions have been imposed on all data values.
Example: If 10 students have mean quiz score 90%, we can freely assign values to 9 scores, but the 10th is then set.
Review: Point Estimation
Definition: Estimation of a population parameter (μ) based on a sample statistic
I.e., given a collection of n sample values, we develop an estimate of the value of the population mean μ
Example: Using statistics from a sample of 106 body temperatures to estimate the population mean μ (parameter) of all body temperatures
Assumptions:
Sample is simple random sample (SRS); i.e., all samples of same size have equal chance of being selected.
If sample size is small (n ≤ 30), there is a loose requirement that the sample is from a normally distributed population.
Example: Use body temperature sample of 106 individuals to find best point estimate of the population mean μ of all body temperatures.
The best point estimate of population mean μ is the sample mean x-.
x- = 98.2°F
The best estimate of population mean μ of all body temperatures is 98.2°F.
Definitions
Confidence interval: Range (or interval) of values used to estimate the true value of a population parameter with a specific level of confidence
Level of confidence: Probability that a population parameter is contained within a particular confidence interval
Meaning: Tells us percentage of times that the confidence interval actually does contain the population parameter.
Notation: 1 − α (note that α is the complement of level of confidence).
Example: If level of confidence is 0.95, α = 0.05. This is the same as being 95% confident.
...
General guidelines for interpreting confidence intervals:
There is a fixed and constant value of μ.
If we use sample data to estimate limits such as 98.08 to 98.32, these limits do or do not enclose the population mean μ.
We cannot determine whether the limits do or do not enclose the population mean μ without knowing the true value of μ.
Because μ is a constant, not a random variable, it is incorrect to say that μ has a 95% chance of falling within these limits.
Either μ will fall within these limits or it will not; there is no probability involved.
Margin of Error (E)
Composed of two parts: A critical value and a standard error
Definition: Maximum likely (with probability 1 − α) difference between observed sample mean x- and true value of population mean μ
Formula: Multiply critical value by standard error (SE), which is the standard deviation of sample means.
E=Critical Value×SE
Where SE=SDn‾‾√/
In considering E, think of the difference as an error.
Probability that a sample mean will be in "error" by no more than E is 1 − α.
Probability that a sample mean will be in "error" by more than E is α.
critical value
each tale is alpha/2

Assume sample means have bell-shaped distribution.
Sample means have relatively small chance (probability = α) of falling in extreme tails, referred to as the critical region.
Denote area of each tail by α/2.
By rule of complements, sample mean will fall between tails with probability 1 − α.
Value separating right tail (or left tail) is referred to as the critical value.
It is on the borderline separating sample means that are likely to occur from those that are unlikely to occur.
Finding Critical Values (Known σ)
When σ is known, the critical value is the z-score on borderline separating sample statistics that are likely to occur from those that are unlikely to occur.
Notation: zα2/
This is the positive z-score separating area of α/2 in right tail of standard normal distribution.
Example: Critical Values (Unknown σ)
We will use the t-distribution to determine the critical value in this case.
Degrees of freedom also comes into play; critical value varies based on this factor.
When σ is unknown, we will use t-distribution to find the number on borderline separating sample statistics likely to occur from those that are unlikely to occur.
Notation: tdf,α2
Positive t separates area of α/2 in right tail of t-distribution; negative t separates area in the left tail.
Confidence Interval for Population Mean μ Revisited
General formula: (x-−E,x-+E)


σ Known σ Unknown
E=(zα2)×(σn‾‾√) E=(tn−1,α2)×(sn‾‾√)
df for CI
n-1
Recommended textbook explanations

Statistical Techniques in Business and Economics

15th EditionDouglas A. Lind, Samuel A. Wathen, William G. Marchal
1,236 explanations

Mathematical Statistics with Applications

7th EditionDennis Wackerly, Richard L. Scheaffer, William Mendenhall
3,341 explanations

Probability

1st EditionJim Pitman
464 explanations

A Survey of Mathematics with Applications

10th EditionAllen R. Angel, Christine D. Abbott, Dennis C. Runde
6,365 explanations