Probability Models
Given that the random variable X represents the number of goals scored by a soccer team in a match, where X has the following probability distribution:
P(X = 0) = 0.2
P(X = 1) = 0.5
P(X = 2) = 0.3
Calculate the expected value, E(X), and variance, Var(X).
E(X) = 0 * 0.2 + 1 * 0.5 + 2 * 0.3 = 1.1
Var(x) = (0 - 1.1)^2 * 0.2 + (1 – 1.1)^2 * 0.5 + (2 – 1.1)^2 * 0.3 = 0.49
The average weight of apples from a farm is normally distributed with a mean of 200 grams and a standard deviation of 15 grams. Using the empirical rule, estimate the percentage of apples that weight between 215 and 245 grams.
15.85%
16% to the right of 215
0.15% to the right of 245
16% - 0.15% = 15.85%
A population has 42% left-handed individuals. A random sample of 200 people is selected.
Describe the sampling distribution of p_hat (shape, mean, standard deviation)
Mean = 0.42
std dev = sqrt((0.42*(1 – 0.42) / 200) = 0.0349
Shape: approximately normal since 200*0.42 = 84 and 200*(1 – 0.42) = 116
Let X represent the number of hours a student studies per week, with E(X) = 15 and Var(X) = 9.
Let Y = 2X + 5
Compute E(Y)
Compute Var(Y)
E(Y) = 2 * 15 + 5 = 35
Var(aX + b) = a2 * Var(X) = 22 * 9 = 36
A factory produces machine parts, and the time to complete a part follows a normal distribution with a mean of 120 minutes and a standard deviation of 10 minutes. Using the empirical rule, what is P(100 <= X <= 110) – P(X >= 150)?
P(100 <= X <= 110) = (95% - 68%) / 2 = 13.5%
P(X >= 150) = 0.15%
13.5% - 0.15% = 13.35%
If you increase the sample size for constructing a confidence interval, what happens to the width of the interval? Explain why.
If you increase the confidence level of a confidence interval, what happens to the width of the interval? Explain why.
As the sample size increases, the width of the confidence interval decreases. The larger the sample size, the more precise the estimate of the population proportion is (also, computationally, a higher sample size → a smaller standard deviation).
As the confidence level increases, the width of the confidence interval increases. To be more confident that the population proportion is within an interval, we would want that interval to have a larger range (also, computationally, a higher confidence interval → a higher critical value).
Imagine you are the line manager at a very large factory. Assume each product is either defective or not defective. The non-defective rate for each product is 88%. What is the probability the first defective product is found after the 12th product?
P(X > 12) = 1 – P(X <= 12)
= 1 – (1 – (1-.12)|12|) = 0.2157
The population of annual incomes of a certain city is normally distributed with a mean of $45,000 and a standard deviation of $12,000.
What is the probability that a randomly selected resident of the city makes more than $58,000?
1 – norm.dist(58000, 45000, 12000, TRUE)
Assume the population proportion of Iowa students who chose Tippie as their favorite study location is 0.65. Further assume we sample 100 students.
What is the probability when sampling 100 students, the probability the proportion of students who chose Tippie is less than 0.38. Report the Excel formula.
Mean = 0.65 std dev = sqrt((0.65*0.35 / 100)) = 0.048
X ~ N(0.65, 0.048) since 100*0.65 > 10 and 100*0.35 > 10
P(X < 0.38) = norm.dist(0.38, 0.65, 0.048, TRUE)
In a game, the probability of getting a perfect score on a level is p = 0.2. Let X be the number of attempts needed to get the first perfect score.
Calculate the expected number of attempts until the first perfect score.
Calculate Var(X).
E(x) = 1 / 0.2 = 5 attempts
Var(X) = (1 – 0.2) / 0.22 = 20
Let X ~ N(15, sigma) and P(X < 18.75) = 0.9985
Sigma = ?
1.25
z = 3 (since X is 3 standard deviations above the mean (shown by the 0.9985 percentile))
3 = (18.75 – 15) / sigma
Sigma = 1.25
A random sample of 200 college students found that 120 students preferred online classes to in-person classes. Construct a 95% confidence interval for the population proportion.
P_hat = 120 / 200 = 0.6 | n = 200
0.60 +/- 1.96*sqrt[(0.6*(1 - 0.6)) / 200]
= [0.532, 0.668]
In a quality control test, a factory has a 90% success rate in producing defect-free products. A batch contains 12 products. Let X represent the number of defect-free products in a batch.
a) What is P(X = 9)?
b) What is E(X)?
c) What is Var(X)?
a) (12 choose 4) * (0.9)9 * (0.1)3 = 0.085
b) 12*0.9 = 10.8
c) 12 * 0.1 * (1 - 0.1) = 1.08
Salaries are normally distributed with a mean of $60,000 and standard deviation of $8,000
What salary marks the top 10%? [Choose to report either the Excel formula or the exact Empirical Rule probability]
*top 10% is the same as the 90th percentile
= norm.inv(0.90, 60000, 8000)
Joey wants to conduct a study over the population proportion of Iowa students who know how to use the Transit app. He wants to create a confidence interval at 90% confidence with a range of 6%. If the sample proportion of Iowa students who don't know how to use the Transit app is 0.82, Joey should sample _____ students.
p_hat = 1 - 0.82 = 0.18 | CV for 95% CI = 1.96 | MoE = 6% / 2 = 3%
N = [0.18 * (1 - 0.18) * 1.962] / 0.032 = 631
*We always have to round up. Also, when you are not given a proportion, you should use 0.5.