Probability and Inferential Statistics
A number you derive from a population
A number you derive from a sample
A survey of the whole population
Probability & Non-Probability Samples
Every case in the population has the same chance of being selected
A specific group is being used as your sample. Surveying students enrolled in a class
We want to know what % of students work during the semester.
We draw a sample of 500 from a list of all students at the university
N = 20,000 (all students at university)
P = 500/20,000
Use a table of random numbers to selected 500 ID numbers with 6 digits
6 digits will be chosen 500 times until they match up with student numbers
After questioning each of these 500 students, we find that 368 (74%) work during the semester.
Population – 20,000
Sample – 500
Statistic – 74%
Parameter – Doesn't directly appear (it's implicit)
(% of all students in the population who held a job)
Variables (e.g., sample mean, sample proportion)
The sample will differ from the population purely by chance
Positive Sampling Error
Making the statistic exceed the population
Negative Sampling Error
Making the statistic less than the population parameter
Sample statistic = population parameter + sampling error
The theoretical, probabilistic distribution of a statistic for all possible samples of a given size (n).
Construction of a Sampling Distribution
Statistic is used to estimate a parameter.
Not all statistics will have the same value.
What is the distribution of the values that we can get for the statistic?
Standard Error = population standard error / square root of the population size
The average age for a population of doctors in a hospital is 51.6 years, What does this mean value represent?
What does it mean for a sample to be representative
The sample reproduces the important characteristics of the population
Which set of symbols represents the standard deviation of the sampling distribution?
Which of these terms is synonymous with the standard error of the mean?
The standard deviation of a sampling distribution
Two Estimation Procedures
A sample statistic used to estimate a population parameter
Consist of a range of values instead of a single point
Example of point estimate:
50% of Canadians drive less because of gas.
Example of confidence:
Between 47% and 53% of Canadian drivers drive less due to high gas prices.
- Point estimate is in the middle
- Lower and upper bound of C.I: 47% and 53%
- Margin of Error: radius or spread of the confidence interval (3%)
Criteria for Choosing Estimators
An estimator is unbiased if the mean of its sampling distribution is equal to the population value of interest
The extent to which the sampling distribution is clustered around its mean
If n is large, we know that the sample mean/proportion is equal to the population parameter and: (image)
Very good (68 out of 100 chances) that our sample outcome is within +/- 1 standard deviation of the true population parameter
Excellent (95 out of 100) that it is within +/- 3 standard deviations
In less than 1% of cases, a sample outcome will lie further away than +/- 3 standard deviations
Getting back to the matter of dispersion: standard error σx̄ (standard deviation of the sampling distribution) = σ/(√n)
Standard error is an inverse function of n: as sample size increases, σx̄ will decrease
The smaller the standard deviation of a sampling distribution, the greater the clustering and the higher the efficiency.
Constructing Confidence Intervals
1. Set the alpha, a
2. Find the Z score (or critical value) associated with alpha
3. Construct the confidence interval (we will substitute values into the appropriate formulas for confidence interval)
Constructing Confidence Intervals - Set the Alpha
1. Alpha = the probability that the interval will be wrong, I.e., it doesn't include the population parameter.
The commonly used alpha level 0.05 corresponds to a 95% confidence level.
If an infinite number of intervals were constructed at the 0.50 alpha level (all other things being equal). 95% of them would contain the population value; 5% would not.
Constructing Confidence Intervals - Find Z Score
For an interval estimate based on +/-1.96 Z's:
The probabilities are that 95% of all such interval will include or overlap the population value
We can be 85% confident that the interval around our one sample outcome contains the population value
Point Estimate +/- Margin of Error
Point Estimate +/- (Critical Value * Standard Error)
The margin of error depends on:
(1) the standard error for statistic AND
(2) a "critical value/Z score" based on the confidence level
Constructing Confidence Intervals for Proportions
Point Estimate +/- (Critical Value/Score) x Standard Error)
for large samples (interval estimation for proportions based on small samples) (n<100) not covered)
What proportion of students at your university missed at least one day of classes because of illness last semester?
Out of a random sample of 200, 60 reported having missed classes: Ps = 60/200 = .30
Confidence Intervals for Means
formula for large samples (n≥100)
You want to estimate the average IQ of a community using a random sample of 200 residents
- with a sample mean IQ of 105
- assuming a population standard deviation for IQ scores of 15
Alpha set at .05 (i.e. we are willing to run a 5% chance of being wrong).
What is the corresponding Z score ?
What is the formula?
Three differences to Formula 6.1:
- σ is replaced by s
- n is replaced by n–1 to correct for the fact that s is a biased estimator of σ
To construct confidence intervals from sample means when s is unknown, we must use a different theoretical distribution, called the Student’s t distribution.
The shape of the t distribution varies as a function of sample size.
- Distribution is a family of curves, each curve is defined by its degrees of freedom – a value indicating the number of scores in a sample that are “free to vary” when calculating statistics.
- Degrees of freedom (df = n–1).
- As n increases, s becomes a more and more reliable estimator of the population standard deviation (σ)
t distribution becomes more and more like the Z distribution.
Smaller samples: t distribution is flatter and has heavier tails than Z distribution.
The Z and t distribution are essentially identical when the sample size is greater than 100.
Find t score for alpha = 0.05 for n=30
Degrees of freedom (df = n-1): 30 – 1 = 29
t score: ±2.045