Due 10/02/2022
1. 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.
Investment A
Profit Probability
$100 1/2 or 0.5
-$100 1/2 or 0.5
Expected Profit: $0
(100 * 0.5) + (-100 * 0.5) = 0
Investment B
Profit Probability
$500 3/10 or 0.3
-$300 7/10 or 0.7
Expected Profit: $-60
(500 * 0.3) + (-300*0.7) = -60
Investment C
Profit Probability
$600 2/10 or 0.2
$-150 8/10 or 0.8
Expected Profit: $0
600 x 0.2 + (-150) x 0.8 =0
Variance: Var(Profit) Variance: Var(Profit) Variance: Var(Profit)
(100-0)²*0.5 + (500-(-60))²*0.3 + (600-0)²*0.2 + (-150-0)2*0.8
(-100-0)²*0.5 = 0
SD = Profit²
√ 0 = $0
Range: [-100,100]
(-300-(-60))²*0.7 = 134400
SD = Profit²
√ 134400 = $366.61
Range: [-300,500]
= 54000
SD = Profit2 = √ 54000
= $232.38
Range: [-150, 600]
b) Which strategy has the highest expected amount of return?
Investment strategies A & C have the highest expected amount of return as they both have an expected profit of $0 which is more than strategy B, which is $-60.
c) Which strategy has the highest variance and standard deviation (the riskiest)?
Investment strategy B has the highest variance and the highest standard deviation.
2. 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?
3. Suppose that total carbohydrate intake in 12- to 14-year old 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-year old boys?4. 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 criterion, 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?
5. Another study is conducted to estimate the average BMI for middle-aged 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 is 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.
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