Question 1199759
**a. Probability of exactly 22 homeowners:**

* This follows a binomial distribution with:
    * n = 35 (sample size)
    * p = 0.62 (probability of success - homeowner)
    * q = 1 - p = 0.38 (probability of failure - not a homeowner)
    * k = 22 (number of successes)

* Using the binomial probability formula:
   P(X = k) = (nCk) * p^k * q^(n-k) 
   where nCk = n! / (k! * (n-k)!)

* P(X = 22) = (35C22) * (0.62)^22 * (0.38)^(35-22) 
   P(X = 22) ≈ 0.0898

**b. Probability of at most 24 homeowners:**

* P(X ≤ 24) = P(X = 0) + P(X = 1) + ... + P(X = 24)

* This requires calculating the binomial probability for each value of X from 0 to 24 and summing them. 

* Using a calculator or statistical software (like Excel or a TI-84 calculator with the binomcdf function):
   P(X ≤ 24) ≈ 0.9588

**c. Probability of at least 22 homeowners:**

* P(X ≥ 22) = P(X = 22) + P(X = 23) + ... + P(X = 35)

* This requires calculating the binomial probability for each value of X from 22 to 35 and summing them.

* Alternatively:
   P(X ≥ 22) = 1 - P(X ≤ 21)

* Using a calculator or statistical software:
   P(X ≥ 22) ≈ 0.2493

**d. Probability of between 17 and 21 homeowners (inclusive):**

* P(17 ≤ X ≤ 21) = P(X = 17) + P(X = 18) + ... + P(X = 21)

* Calculate each probability using the binomial probability formula and sum them up.

* Using a calculator or statistical software:
   P(17 ≤ X ≤ 21) ≈ 0.6550

**In summary:**

* a) Probability of exactly 22 homeowners: 0.0898
* b) Probability of at most 24 homeowners: 0.9588
* c) Probability of at least 22 homeowners: 0.2493
* d) Probability of between 17 and 21 homeowners: 0.6550