1
STATS 2110 MATHEMATICS ACTUAL
EXAM WITH ACTUAL NOTES SPECIFIC
TO REAL EXAM QS AND AS
,2
Final Exam STATS 2120 Notes:
Date: Notes:
3/31 One-sample z-test for proportions:
Review:
● Statistical Inference provides methods for drawing conclusions about a population based on the sample data
● Two of the most common types of statistical inference are:
○ Confidence intervals: estimate the value of a population parameter using sample statistics
○ Significance testing: assess a claim made about the value of a population parameter using evidence from sample
statistics
● Confidence intervals take the form:
○ Estimate +/- margin of error
○ = estimate +/- (critical value * standard deviation of estimate)
● We choose the critical value such that our interval will be accurate(include the parameter value) for C% of all possible
samples
Categorical Variables:
● Examples:
○ In the last week, have you smoked a cigarette?
■ Yes
■ No
○ How satisfied are you with the condition of your housing?
■ Very unsatisfied
■ Unsatisfied
■ Neutral
■ Satisfied
■ Very satisfied
● We are often interested in a population proportion(p) to describe the distribution of a categorical variable
○ Examples
■ The proportion of people who smoked in the last 7 day
■ The proportion of people who are dissatisfied or very dissatisfied with the condition of their housing
● Each individual in the population, and hence each individual in the sample, either has the characteristic of interest (success)
or does not have that characteristic (failure)
, 3
○ Example: The proportion of people who are dissatisfied or very dissatisfied with the condition of their housing
■ Success: dissatisfied , or very dissatisfied
■ Failure: neutral, satisfied, or very satisfied
Population Proportion
● Let's define a variable X in this way:
○ Xi = {1 if a success is observed | 0 if a failure is observed
● What is the probability distribution of X?
○
Outcome Probability
0(failure) 1-p
1(success) p
● What is the meaning of X, μx?
○ μ = Σ xi * pi
= 0 * (1 - p) + 1 * p
=0+p
=p
● What is the variance of X, σ2x?
○ σ2 = Σ (xi - μi)^2 *Pi
= (0 - p)^2 * (1-p) +(1-p)^2 * p
= p^2 * (1-p) + (1-p)^2 * p
= p^2 - p^3 + (1 - 2p + p^2) * p
= p^2 -P^3 +p - 2p^2 + p^3
= p-p^2
= p * (1 - p)
Sample Proportion
● We can estimate the population proportion from sample data using the sample proportion, p^
○ p^ = #successes in the sample / n
● We can define the sample proportion using our variable X:
𝑛
○ −
p^ = #successes in the sample / n = ( ∑ 𝑥𝑖)/𝑛 = 𝑥
𝑖=1
Sampling distribution of sample proportion
STATS 2110 MATHEMATICS ACTUAL
EXAM WITH ACTUAL NOTES SPECIFIC
TO REAL EXAM QS AND AS
,2
Final Exam STATS 2120 Notes:
Date: Notes:
3/31 One-sample z-test for proportions:
Review:
● Statistical Inference provides methods for drawing conclusions about a population based on the sample data
● Two of the most common types of statistical inference are:
○ Confidence intervals: estimate the value of a population parameter using sample statistics
○ Significance testing: assess a claim made about the value of a population parameter using evidence from sample
statistics
● Confidence intervals take the form:
○ Estimate +/- margin of error
○ = estimate +/- (critical value * standard deviation of estimate)
● We choose the critical value such that our interval will be accurate(include the parameter value) for C% of all possible
samples
Categorical Variables:
● Examples:
○ In the last week, have you smoked a cigarette?
■ Yes
■ No
○ How satisfied are you with the condition of your housing?
■ Very unsatisfied
■ Unsatisfied
■ Neutral
■ Satisfied
■ Very satisfied
● We are often interested in a population proportion(p) to describe the distribution of a categorical variable
○ Examples
■ The proportion of people who smoked in the last 7 day
■ The proportion of people who are dissatisfied or very dissatisfied with the condition of their housing
● Each individual in the population, and hence each individual in the sample, either has the characteristic of interest (success)
or does not have that characteristic (failure)
, 3
○ Example: The proportion of people who are dissatisfied or very dissatisfied with the condition of their housing
■ Success: dissatisfied , or very dissatisfied
■ Failure: neutral, satisfied, or very satisfied
Population Proportion
● Let's define a variable X in this way:
○ Xi = {1 if a success is observed | 0 if a failure is observed
● What is the probability distribution of X?
○
Outcome Probability
0(failure) 1-p
1(success) p
● What is the meaning of X, μx?
○ μ = Σ xi * pi
= 0 * (1 - p) + 1 * p
=0+p
=p
● What is the variance of X, σ2x?
○ σ2 = Σ (xi - μi)^2 *Pi
= (0 - p)^2 * (1-p) +(1-p)^2 * p
= p^2 * (1-p) + (1-p)^2 * p
= p^2 - p^3 + (1 - 2p + p^2) * p
= p^2 -P^3 +p - 2p^2 + p^3
= p-p^2
= p * (1 - p)
Sample Proportion
● We can estimate the population proportion from sample data using the sample proportion, p^
○ p^ = #successes in the sample / n
● We can define the sample proportion using our variable X:
𝑛
○ −
p^ = #successes in the sample / n = ( ∑ 𝑥𝑖)/𝑛 = 𝑥
𝑖=1
Sampling distribution of sample proportion
