Mid term Equations
Suppose that a disease is inherited via a dominant mode of inheritance and that only one of the two parents is affected with the disease. The implications of this mode of inheritance are that the probability is 0.3 that any particular offspring will get the disease, and is 0.15 that all the offspring will get the disease. For a family with two children, treat two affected children as two different events such that O = {older child is affected}, Y = {younger child is affected}.
(a) Are the events O = {older child is affected} and Y = {younger child is affected} independent? Why or why not? (b) What is the probability that at least one sibling is affected? (c) Given the younger child is affected, what is the probability that the older child is affected?
Suppose an influenza epidemic strikes a city. In 10% of families the mother has influenza; in 5% of families the father has influenzas. Suppose the events M = {mother has influenza} and F = {father has influenza} are independent.
(a) What is the probability that in a family both father and mother will have influenza? (b) What is the probability that exactly one of the parents will get influenza? (c) What is the probability that neither of the parents will get influenza? (d) What is the conditional probability that the mother will have influenza given that the father has influenza?
Consider the typhoid-fever example. Suppose the number of deaths from typhoid fever over a 1-year period is Poisson distributed with parameter μ = 4.6 (average deaths over a year).
What is the probability of exact 3 deaths over a year?
FIU students' weight follows a normal distribution with a mean avg = 150 lbs, a variance = 100 lbs2, and a standard deviation sig = 10 lbs. Such a distribution is often indicated by the symbols N(150, 100).
What's the probability that the weight of a random selected student will be more than 170 lbs? What's the probability that the mean weight of 16 random selected students will be more than 170 lbs?
. Another study is conducted to estimate the average BMI for middle-age US females. The study includes a random sample of 15 middle-aged US females with a sample mean of 22.5 and a sample standard deviation of 3.
1) Is it safe to assume the sampling distribution of BMI for 15 randomly selected females approximately normal? Why or why not? 2) Use the appropriate formula to construct a 95% CI for the true mean BMI for middle-aged US females.
Let E be the event that a student is taking an epidemiology course, and B be the event that a student is taking a biostatistics course. Assume P(E) = 0.30, P(B)= 0.25, and P( ) = 0.10. Find the probability that a student is taking:
1.At least one of courses of epidemiology or biostatistics. 2.Neither epidemiology nor biostatistics. 3.Biostatistics but not epidemiology.
You must choose one of the coin-matching games to play. Which do you prefer? Assume you toss up two fair coins.
1.Game 1: You win $100 if both faces match, lose $100 if they do not match. 2.Game 2: You win $100 if the match is tails, and $200 if the match is heads; lose $100 if they do not match.
Let E be the event that a student is taking a epidemiology course, and B be the event that a student is taking a biostatistics course. Assume P(E) = 0.30, P(B)= 0.25, and P( ) = 0.10. Find the probability that:
A student is taking epidemiology given that he/she is taking biostatistics. A student is taking biostatistics given that he/she is taking epidemiology.
Let E be the event that a student is taking a epidemiology course, and B be the event that a student is taking a biostatistics course. Assume P(E) = 0.30, P(B)= 0.25, and P( ) = 0.10.
Are taking epidemiology and taking biostatistics mutually exclusive? Are taking epidemiology and taking biostatistics independent?
Body Mass Index (BMI) is a number calculated from a person's weight and height and is calculated by the formula: weight (kg) / [height (m)]2. Though somewhat controversial and not meant to be used as a diagnostic criteria, it can be used to characterize body type and (with caution) fitness and health using readily available data. Commonly accepted classifications are listed below: • Below 18.5 is underweight; • 18.5 to 25 is normal; • 25 to 30 is overweight; • Above 30 is obese.
Based on a study, the BMI for middle-aged US males has a normal distribution with a mean of 23.5 and a standard deviation of 4. What is the probability that the average BMI is considered underweight based on BMI criteria for a randomly selected sample including 50 middle-aged US males?
Find the 20th percentile of a standard normal distribution.
Find the 20th percentile of N(10, 25)
Suppose that 1 woman in 10 with positive mammograms will be diagnosed with breast cancer within 2 years, and 7% of the general population of women will have a positive mammogram. What is the probability for a woman with positive mammograms and developing breast cancer over next 2 years?
Fricck around and find out
Suppose that among 100,000 women with negative mammograms 20 women will be diagnosed with breast cancer within 2 years, whereas 1 woman in 10 with positive mammograms will be diagnosed with breast cancer within 2 years. And 7% of the general population of women will have a positive mammogram. What is the probability for developing breast cancer over next 2 years among women in the general population?
Frick around and find out
Experiment: Roll one 6-sided die. Observe the face. Sample space:S={1, 2, 3, 4, 5, 6}.
Let A be the event I roll an even number. A= Let B be the event I roll a number greater than B= Let C be the event I roll a number 0. C=
Let X ~ N(5, 100).
P(X ≤ 0) = What's the probability within one sd from mean? P(5-10 ≤ X ≤ 5+10)=
The table on Canvas shows the standard normal cumulative probabilities F(z)=P(Z ≤ z) for many values of z. Always draw a curve! Here are some examples:
P(Z ≤ -0.5)= P(Z ≥ 0.5)= P(-1 ≤ Z ≤ 1)=
In a survey, 28 student responded to the question "If you have an Instagram account, how many photos did you post in the past week?" The above is the frequency table. What is the probability that a randomly selected student posted exactly one photo in the past week?
Photo 1.
What is the probability that a random selected student posted no more than one photo? (What is the event?) What is the probability that a random selected student posted at least one photo? (What is the event?)
Photo 1.
Consider the random variable X = Apgar Score. Compute the mean of the random variable X and interpret it in context. The mean Apgar score of a randomly selected newborn is _____. This is the long-term average Apgar score of many, many randomly chosen babies.
Photo 2
The probability distribution for X = Apgar scores is shown below: a.Show that the probability distribution for X is legitimate. b.Make a histogram of the probability distribution. Describe what you see. c.Apgar scores of 7 or higher indicate a healthy baby. What is P(X≥ 7)?
Photo 2
Probability Distribution
Toss one coin two times. Count number of heads.
Consider the following investment strategies. A: 50% chance of winning $100, and 50% chance of losing $100. B: 30% chance of winning $500, and 70% chance of losing $300. C: 20% chance of winning $600, and 80% chance of losing $150.
a) For each strategy, find the probability distribution of the amount of return. b) Which strategy has the highest expected amount of return? c) Which strategy has the highest variance and standard deviation (the riskiest)?
Suppose that total carbohydrate intake in 12- to 14-yearold boys is normally distributed, with mean = 150 g/1000 cal and standard deviation = 20 g/1000 cal.
a) What is the probability that a randomly selected boy in this age range has carbohydrate intake above 190 g/1000 cal? b) What is the probability that a randomly selected boy in this age range has carbohydrate intake below 100 g/1000 cal? c) What is the probability that a randomly selected boy in this age range has carbohydrate intake between 130 and 170 g/1000 cal? d) What is the 85th percentile of the total carbohydrate intake in 12- to 14-yearold boys?
A study considered risk factors for HIV infection among intravenous drug users. It found that 20% of users who had ≤ 100 injections per month (light users) were HIV positive.
a) What is the probability that exactly 4 out of 10 light users are HIV positive? b) What is the probability that at least 4 out of 10 light users are HIV positive?
So 95%=P(? <Z< ? )
do it
Experiment: Toss 1 coin 6 times in a row. Note number of tails. What's the probability of no more than 3 tails?
do it again
Experiment: Toss 1 coin 6 times in a row. Note number of tails. What's the probability of 3 tails?
do it!
What is the value z so that P(-z<Z<z)=95%?
do it!
For = 0.05 for a random sample of size n = 5 from the population of body weights, we had
n= 5 sig = 10 x/ = 153.0