A particular telephone number is used to receive both voice calls and fax messages. Suppose that 40% of the incoming calls involve fax messages, and consider a sample of 15 incoming calls.

A button hyperlink to the SALT program that reads: Use SALT.

(a)

What is the expected number of calls among the 15 that involve a fax message?

E(X) =

(b)

What is the standard deviation of the number among the 15 calls that involve a fax message? (Round your answer to three decimal places.)

(c)

What is the probability that the number of calls among the 15 that involve a fax transmission exceeds the expected number by more than 2 standard deviations? (Round your answer to three decimal places.)

Let's break down the problem step by step.

### Given:
- Probability of an incoming call involving a fax: \( p = 0.40 \)
- Sample size: \( n = 15 \)
- Number of calls is modeled using a **Binomial Distribution**: \( X \sim \text{Binomial}(n = 15, p = 0.40) \)

#### Part (a) - Expected Value
The expected number of calls involving a fax is given by the formula for the expected value of a binomial distribution:
\[
E(X) = n \cdot p
\]
Substitute the values of \( n \) and \( p \):
\[
E(X) = 15 \cdot 0.40 = 6
\]

So, the expected number of fax-related calls is:
\[
E(X) = 6
\]

#### Part (b) - Standard Deviation
The standard deviation for a binomial distribution is given by:
\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)}
\]
Substitute the values:
\[
\sigma = \sqrt{15 \cdot 0.40 \cdot (1 - 0.40)} = \sqrt{15 \cdot 0.40 \cdot 0.60}
\]
Let's calculate that.

\[
\sigma = \sqrt{15 \cdot 0.40 \cdot 0.60} = \sqrt{3.6} \approx 1.897
\]

So, the standard deviation is approximately:
\[
\sigma \approx 1.897
\]

#### Part (c) - Probability of Exceeding the Expected Number by More than 2 Standard Deviations
We need to find the probability that the number of calls exceeds the expected value by more than 2 standard deviations. This means we are interested in:

\[
P(X > E(X) + 2\sigma)
\]

Using the values from parts (a) and (b):
\[
E(X) + 2\sigma = 6 + 2 \times 1.897 \approx 6 + 3.794 = 9.794
\]

So we want to calculate \( P(X > 9.794) \), which is the probability that more than 9.794 calls involve a fax. Since \( X \) is discrete, this is equivalent to finding \( P(X \geq 10) \).

We will calculate \( P(X \geq 10) \) using the binomial distribution formula or a binomial calculator.

Let me calculate that for you.

The probability that the number of calls involving a fax exceeds the expected number by more than 2 standard deviations is approximately:

\[
P(X \geq 10) \approx 0.034
\]

### Summary of Results:
- **(a)** Expected number of fax-related calls: \( E(X) = 6 \)
- **(b)** Standard deviation: \( \sigma \approx 1.897 \)
- **(c)** Probability that the number of calls exceeds the expected number by more than 2 standard deviations: \( P(X \geq 10) \approx 0.034 \)

Do you want more details or have any other questions?

Here are 5 related questions to explore:
1. How would the expected number change if the probability of a fax call was 60%?
2. What is the probability that exactly 8 calls involve a fax transmission?
3. What is the cumulative probability that at most 5 calls involve a fax transmission?
4. How would the standard deviation change for a sample of 25 calls?
5. How can we approximate the binomial distribution using a normal distribution for large \(n\)?

**Tip:** The binomial distribution is symmetric when \( p = 0.5 \), but becomes skewed when \( p \) is far from 0.5.

A particular telephone number is used to receive both voice calls and fax messages. Suppose that 40% of the incoming calls involve fax messages, and consider a sample of 15 incoming calls.
(a) What is the expected number of calls among the 15 that involve a fax message?
(b) What is the standard deviation of the number among the 15 calls that involve a fax message? (Round your answer to three decimal places.)
(c) What is the probability that the number of calls among the 15 that involve a fax transmission exceeds the expected number by more than 2 standard deviations?

Explore how to calculate the expected number, standard deviation, and probability of fax transmissions using a binomial distribution. In this problem, 40% of calls involve fax messages, with a sample size of 15 calls. Learn how to apply key formulas and theorems. Suitable for Grades 11-12.

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