In a study by Peter D. Hart Research Associates for the Nasdaq Stock Market, it was determined that 20% of all stock investors are retired people. In addition, 40% of all adults invest in mutual funds. Suppose a random sample of 25 stock investors is taken.
a. What is the probability that exactly seven are retired people?
b. What is the probability that 10 or more are retired people?
c. How many retired people would you expect to find in a random sample of 25 stock investors?
d. Suppose a random sample of 20 adults is taken. What is the probability that exactly seven adults invested in mutual funds?
e. Suppose a random sample of 20 adults is taken. What is the probability that fewer than six adults invested in mutual funds?
f. Suppose a random sample of 20 adults is taken. What is the probability that none of the adults invested in mutual funds?
g. Suppose a random sample of 20 adults is taken. What is the probability that 12 or more adults invested in mutual funds?
h. For parts e�g, what exact number of adults would produce the highest probability? How does this compare to the expected number
Let X be the random variable denoting the number of retired people
a) P(X=7) = 25C7∗0.27∗0.818 = 0.1108
b) P(X>=10) =1-P(X<=9) Using the Binomdist() excel Function , which calculates P(X<=x) =1-.98266 = 0.0173
c)In a random sample of 25, the expected number is : 25*0.2 =5 Let Y be the random variable denoting number ofU.S. adults investing in mutual funds. Y~Bin(20,0.4)
To solve these problems, we will use the binomial probability formula. The formula is:
P(X = k) = (n choose k) * p^k * q^(n-k)
Where:
- P(X = k) represents the probability of getting exactly k successes in n trials.
- (n choose k) is the binomial coefficient, which calculates the number of ways to choose k items from a set of n items.
- p is the probability of success in a single trial.
- q is the probability of failure in a single trial, which is calculated as 1 - p.
- k is the number of successes.
- n is the total number of trials.
Now let's solve each part step by step:
a. What is the probability that exactly seven are retired people?
In this case, we have n = 25 (total number of stock investors) and p = 0.20 (probability of a stock investor being retired).
Using the formula, the probability is:
P(X = 7) = (25 choose 7) * (0.20)^7 * (0.80)^(25-7)
b. What is the probability that 10 or more are retired people?
To calculate this, we need to sum the probabilities of getting 10, 11, 12, ..., 25 retired people.
P(X ≥ 10) = P(X = 10) + P(X = 11) + ... + P(X = 25)
c. How many retired people would you expect to find in a random sample of 25 stock investors?
The expected number of retired people is given by the formula:
Expected number = n * p
d. Suppose a random sample of 20 adults is taken. What is the probability that exactly seven adults invested in mutual funds?
In this case, we have n = 20 (total number of adults) and p = 0.40 (probability of an adult investing in mutual funds).
Using the binomial formula, calculate P(X = 7).
e. Suppose a random sample of 20 adults is taken. What is the probability that fewer than six adults invested in mutual funds?
To solve this, we need to find P(X < 6), which is the sum of P(X = 0), P(X = 1), P(X = 2), P(X = 3), P(X = 4), and P(X = 5).
f. Suppose a random sample of 20 adults is taken. What is the probability that none of the adults invested in mutual funds?
To calculate this, we need to find P(X = 0).
g. Suppose a random sample of 20 adults is taken. What is the probability that 12 or more adults invested in mutual funds?
To find this probability, we need to sum the probabilities of getting 12, 13, ..., 20 adults who invested in mutual funds.
h. For parts e-g, what exact number of adults would produce the highest probability? How does this compare to the expected number?
To find the exact number of adults that would produce the highest probability, calculate the probabilities for different numbers and compare them. The number of adults that corresponds to the highest probability will be the answer. Compare this number to the expected number calculated in part c.