Answer each question to the best of your ability.  If not otherwise indicated, round your only final answers (not the middle steps) to two decimal places. The National Health Interview Survey reports that 41% of the telephone users no longer use landlines, and have switched completely to cell phone use. Suppose we take random sample of 10 telephone users. Use the table below to answer the question a.

X	P = 0.41
0 	0.0051
1 	0.0355
2 	0.1111
3 	0.2058
4 	0.2503
5 	0.2087
6 	0.1209
7 	0.0480
8 	0.0125
9 	0.0019
10 	0.01

a)	Find the probability that the sample contains at least 5 users who have abandoned their landlines? Show all your steps. (5 points)
Hint P(X at least 5) = P( X= 5)+ P( X= 6)+ P( X= 7)+ P( X= 8)+ P( X= 9)+ P( X= 10)

b) Assume that the National Health Interview Survey reports that 44% of telephone users no longer use landlines and have switched completely cell phone use. Suppose we take a random sample of 10 telephone users. Find the probability that the sample contains exactly 5 users who have abandoned their landlines? Show all your steps. (5 points)
Hint use the binomial probability formula given on page 203 in the textbook.

Let's tackle each part of the question step by step.

### Part (a): Find the probability that the sample contains at least 5 users who have abandoned their landlines.

We are given that \( P(X \geq 5) \) is the probability that at least 5 users have abandoned their landlines. According to the problem, we can calculate this by summing the probabilities for \( X = 5 \) to \( X = 10 \).

Given probabilities:

\[
P(X = 5) = 0.2087
\]
\[
P(X = 6) = 0.1209
\]
\[
P(X = 7) = 0.0480
\]
\[
P(X = 8) = 0.0125
\]
\[
P(X = 9) = 0.0019
\]
\[
P(X = 10) = 0.01
\]

Now, we calculate:

\[
P(X \geq 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
\]

\[
P(X \geq 5) = 0.2087 + 0.1209 + 0.0480 + 0.0125 + 0.0019 + 0.01
\]

\[
P(X \geq 5) = 0.4020
\]

So, the probability that the sample contains at least 5 users who have abandoned their landlines is **0.40**.

### Part (b): Find the probability that the sample contains exactly 5 users who have abandoned their landlines (using the binomial probability formula).

We are given:

- \( n = 10 \) (the sample size)
- \( p = 0.44 \) (the probability that a user has abandoned their landline)
- \( k = 5 \) (we are interested in exactly 5 users)

The binomial probability formula is:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

Where:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

First, calculate the binomial coefficient:

\[
\binom{10}{5} = \frac{10!}{5!5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252
\]

Now, calculate the probability:

\[
P(X = 5) = 252 \times (0.44)^5 \times (1 - 0.44)^{10-5}
\]

\[
P(X = 5) = 252 \times (0.44)^5 \times (0.56)^5
\]

\[
P(X = 5) = 252 \times 0.0169 \times 0.0559
\]

\[
P(X = 5) \approx 252 \times 0.000945
\]

\[
P(X = 5) \approx 0.238
\]

So, the probability that the sample contains exactly 5 users who have abandoned their landlines is **0.24**.

---

### Any Questions?

Would you like to go into more details or have any questions? Here are a few follow-up questions that you might find helpful:

1. What is the binomial distribution, and how does it apply in real-world scenarios?
2. How can you use the binomial coefficient in other probability problems?
3. How would the probability change if the sample size increased to 20?
4. What are other methods for solving similar probability problems, like using a cumulative distribution function?
5. Can you explain how the change in \( p \) from 0.41 to 0.44 affected the probability in part (b)?

---

### Tip:
When solving probability problems, always double-check the interpretation of "at least" and "exactly" to ensure you're calculating the correct probabilities.

Calculate the probability of telephone users abandoning landlines based on National Health Interview Survey data. Learn how to compute probabilities for at least 5 and exactly 5 users using binomial distribution. Suitable for Grades 10-12.

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