Probability and Statistics
Measure the covariance and the correlation between two random variables: the temperature outside and the height of a person, and interpret the results.
μ is 100 grams. An employee believes the mean is not 100 grams, forming the alternative hypothesis. Determine whether to use a one-tailed or two-tailed test for hypothesis testing.
Using the critical value approach to hypothesis testing, given an alpha level of 0.05, find the critical value on the standardized normal distribution curve where 5% of the area is in the tail.
Define a random variable capital X where it is equal to 1 if heads and 0 if tails when flipping a fair coin.
Define another random variable capital Y as the sum of the upward face after rolling 7 dice.
Hugo plans to buy packs of baseball cards until he gets the card of his favorite player, but he only has enough money to buy at most four packs. Suppose that each pack has a probability of 0.2 of containing the card that Hugo is hoping for. Let the random variable X be the number of packs of cards Hugo buys. Find the indicated probability: what is the probability that X is greater than or equal to two?
A statistics class has six students with the ages [18, 18, 19, 20, 20, 21]. Construct a sampling distribution of the mean of age for samples with the size of two.
You are planning a full day nature trip for 50 men. The average male drinks 2 liters of water when active outdoors with a standard deviation of 0.7 liters. You will bring 110 liters of water. What is the probability that you will run out of water?
Find the sample size required to estimate the population mean to within 1.5 units where the population standard deviation is 8 at 95% confidence level.
To estimate the population mean to within 1.5 units with 90% confidence, determine the minimum sample size required given the z-critical value is 1.645.
Estimate the average population mean of candy bar lengths from a factory by sampling 100 candy bars and finding a confidence interval for the mean length.
Calculate the expected value of winning a game where if she wins, she receives 100. The probability of winning the game is 20%.
Company XYZ generates a profit of 500 for each defective laptop. If 3 out of every 100 laptops produced are defective, calculate the expected value of profit per laptop.
Given the probability distribution for the random variable x, representing the number of workouts in a week, find the expected value (mean) of x.
What is the probability that the random variables X and Y simultaneously take the values 1 and 3?
For a given joint PMF of three random variables X, Y, and Z, determine the probability that X takes on a specific value. Consider all possible triples where random variable X indeed takes that value and sum over all possible values of Y's and Z's that go together with this particular X.
Determine the PMF of a random variable which is a function of two other random variables X and Y, by finding the probability that the function of X and Y takes on a specific numerical value.
Solve a problem involving a joint probability distribution given continuous random variables and spatial data, similar to analyzing the probability density of a basketball player's position on a court.
Given these sample data, conduct an F-test to determine if the population variances are significantly different at \alpha = 0.05.
Is there a difference between the variances of the number of weeks on the bestseller list for nonfiction and fiction books?
15 New York Times bestselling fiction books had a standard deviation of 6.17 weeks on the list. 16 New York Times best-selling nonfiction books had a standard deviation of 13.12 weeks.
At the 10% significance level, can we conclude there is a difference in the variances?