Probability
Screening Tests
Distributions
Confidence Intervals
Misc.
100
If P(A)=0.2, P(B)=0.3, and P(A ∩ B)= 0.006, are these events independent?
No: if P(A ∩ B)= 0.06, then it would be independent.
Independence = if P(A)*P(B)= P(A ∩ B)
100
In a screening test for colon cancer, it was found that the test had a 0.015 probability of resulting positively for those without colon cancer.
The probability of a negative test result among those who do have colon cancer was 0.881.
Find the specificity of the test
P(T-|Dc) = 1 - P(T+|Dc) -> Specificity = 1 - false positive
= 1 - 0.015 = 0.985
100
Suppose that the average number of pedestrian and bicyclist traffic deaths per month is 2.25 in Iowa.
What is the probability that at least four traffic deaths of these types will occur in a given month?
X ~ Poi(2.25)
P(x ≥ 4) = 1- P(x=0) - P(x=1) - P(x=2) - P(x=3)
=0.1906
100
What is the critical value for a 95% confidence interval?
1.96
100
For mutually exclusive events, what is the probability of A and B occurring
0, they cannot occur at the same time
200
Find the probability that a person is female or prefers going hiking on mountain peaks
P(F) = 45/100
P(M) = 25/100
P(F and M) = 11/100
P(F or P) = (45 +25 - 11)/100 = 59/100 = .59
200
Define and give the notation for the positive and negative predictive values
NPV: P(Dc|T-) = The probability that the individual does not have the disease given that the individual had a negative test result
PPV: P(D|T+) = The probability that the individual does have the disease given that the individual had a positive test result
200
Suppose that Cameron has an 0.82 probability of making a free throw on any one try. Assuming that this probability does not change, out of 7 free throws, how many should we expect Cameron to make?
X ~ Bin(7,0.82)
E(x)=np = 5.75 ~ 6 free throws
200
What is the general form for a confidence interval?
Point Estimate +/- Margin of Error
200
If events are mutually exclusive, what is the probability of A or B occurring?
= P(A) + P(B), there is no intersection to subtract
300
Out of males, what is the probability that the cyclist prefers a hilly path?
P(H|M) = 52/90 = 0.5778
300
In a screening test for colon cancer, it was found that the test had a 0.015 probability of resulting positively for those without colon cancer.
The probability of a negative test result among those who do have colon cancer was 0.881.
Find the sensitivity of the test
P(T+|D) = 1 - P(T-|D) -> sensitivity = 1 - false negative
= 1- 0.881 = 0.119
300
Suppose that the average number of pedestrian and bicyclist traffic deaths per month is 2.25 in Iowa.
What is the probability that there is more than 1 traffic death of these types?
X ~ Poi(2.24)
P(X ≥ 1) = 1 - P(x=0) - P(x=1) = 0.8946
300
Construct a 95% confidence interval for the following:
The average amount of sleep for 81 coffee drinkers is measured. The sample mean is 36 hours a week. Assume the population standard deviation hours of sleep a week is 2 hours a week.
CI 95% = (35.6, 36.4) hours of sleep a week
300
If X ~ Norm(80,0.94^2)
What is the probability that x=6
P(x=6) = 0
The normal distribution is continuous, not discrete. We can only find the area under a given curve
400
What is the probability that a person had no violations in the last year and was a car phone user?
P( no viol ∩ car phone user) = 280/755 = 0.3709
400
In a screening test for colon cancer, it was found that the test had a 0.015 probability of resulting positively for those without colon cancer.
The probability of a negative test result among those who do have colon cancer was 0.881.
Find the positive predictive value of the test if we assume that 25% of the people in this location have colon cancer.
PPV = 0.2484
400
Suppose that Cameron has an 0.82 probability of making a free throw on any one try. Assuming that this probability does not change, find the probability that Cameron makes more than 4 of the next 7 free throws.
X ~ Bin(7,0.82)
P(x>4) = P(x=5) + P(x=6) + P(x=7)
= 0.88459
400
What is the difference between the standard error and the margin of error?
Standard error = standard deviation of a statistic = sigma/sqrt(n)
The margin of error = confidence level * standard error of x bar
400
Name the term:
If the sample size is ‘large,’ the sampling distribution of ̄Xis approximately normal, regardless of the characteristics of the underlying population.
The Central Limit Theorem
500
Are the events "being male" and preferring the hilly path" independent events?
No
P(M) = 90/200 = 0.45
P(hilly)= 90/200 = 0.45
P(M ∩ Hilly) = 52/200 = 0.26
0.2045 ≠ 0.26
other way
500
Define and give the notation for sensitivity and specificity
Sensitivity: P(T+|D) = the probability of obtaining a positive test result given that the individual has the disease
Specificity: P(T-|Dc) = the probability of obtaining a negative test result given that the individual does not have the disease
500
List 1 similarity and 1 difference between the Binomial and Poisson distributions
1. They both events to be independent; They are both discrete functions
Binomial: 2 outcomes, p remains the same, # trials fixed
Poisson: periodically over time, proportional, infinite events possible
500
Construct a 95% confidence interval for the following:
In a population of 5,000 people, the average amount of hours standing a day is measured. A sample of 32 people is taken with a sample mean of standing for 9 hours a day. Assume the population standard deviation hours of standing is 3 hours.
CI 95% = (7.96, 10.04) hours of standing a days
500
Given the sensitivity and the specificity of a screening test, along with the prevalence, the positive predictive value and/or the negative predictive value can be found using ______.
Bayes Theorem