Question 1209735
Here's how to solve this binomial probability problem:

**Understanding the Problem**

This is a binomial probability problem because:

* There are a fixed number of trials (n = 37 Americans).
* Each trial is independent.
* There are only two outcomes (homeowner or not).
* The probability of success (being a homeowner) is constant (p = 0.69).

**Calculations**

The binomial probability formula is:

P(x) = (nCx) * p^x * (1-p)^(n-x)

Where:

* P(x) is the probability of exactly x successes
* n is the number of trials
* x is the number of successes
* p is the probability of success
* nCx is the binomial coefficient, calculated as n! / (x! * (n-x)!)

**(a) Exactly 27 Homeowners:**

P(27) = (37C27) * (0.69)^27 * (0.31)^10
P(27) ≈ 4,262,949,390 * 0.00115 * 0.0000028 ≈ 0.0137

**(b) At Most 27 Homeowners:**

This means 0 to 27 homeowners.  We need to sum the probabilities from P(0) to P(27). It's easier to use the complement rule:

P(x ≤ 27) = 1 - P(x > 27) = 1 - [P(28) + P(29) + ... + P(37)]

Calculating all these probabilities and summing them is tedious.  A binomial cumulative distribution function (CDF) calculator or statistical software is highly recommended for this.  Using such a tool, we get:

P(x ≤ 27) ≈ 0.986

**(c) At Least 23 Homeowners:**

This means 23 to 37 homeowners. Again, it's easier to use the complement:

P(x ≥ 23) = 1 - P(x < 23) = 1 - [P(0) + P(1) + ... + P(22)]

Use a binomial CDF calculator or software:

P(x ≥ 23) ≈ 0.865

**(d) Between 23 and 28 Homeowners (inclusive):**

This means 23, 24, 25, 26, 27, and 28 homeowners.  We can calculate this directly or use the CDF:

P(23 ≤ x ≤ 28) = P(x ≤ 28) - P(x ≤ 22)

Use a binomial CDF calculator or software:

P(23 ≤ x ≤ 28) ≈ 0.992-0.126=0.866


**Summary of Answers (using a calculator/software for parts b, c, and d):**

* (a) P(x = 27) ≈ 0.0137
* (b) P(x ≤ 27) ≈ 0.986
* (c) P(x ≥ 23) ≈ 0.865
* (d) P(23 ≤ x ≤ 28) ≈ 0.866