Let's break down the problem you shared:

Given:

• 8% of visitors to the website are looking for the company’s website (i.e., potential or actual customers).
• A random sample of 40 visitors is taken from the total visitors.
• We are tasked to estimate the following:

(a) The expected number of visitors in the sample who are looking for the company’s website.

The number of visitors who are looking for the company’s website follows a binomial distribution because:

• There are a fixed number of trials (40 visitors).
• Each visitor is either looking for the company’s website or not.
• The probability of success (visitor looking for the company’s website) is 8%, or 0.08.

The expectation $E(X)$ for a binomial distribution $X \sim \text{Binomial}(n, p)$ is given by:

$E(X) = n \times p$

where:

• $n = 40$ (the number of visitors sampled),
• $p = 0.08$ (the probability of a visitor being a customer).

$E(X) = 40 \times 0.08 = 3.2$

So, the expected number of visitors in the sample looking for the company’s website is 3.2.

(b) The standard deviation of the distribution.

The standard deviation $\sigma$ of a binomial distribution is given by the formula:

$\sigma = \sqrt{n \times p \times (1 - p)}$

Substituting the values:

$\sigma = \sqrt{40 \times 0.08 \times (1 - 0.08)} = \sqrt{40 \times 0.08 \times 0.92}$

$\sigma = \sqrt{2.944} \approx 1.715$

Thus, the standard deviation is approximately 1.715 (rounded to three decimal places).

Would you like more details or have any questions?

Here are 5 related questions to expand on this problem:

1. What is the probability of exactly 4 visitors in the sample being actual customers?
2. How would the expected value change if the probability of being a customer increased to 10%?
3. What would the variance be for this distribution?
4. How does increasing the sample size affect the standard deviation of the estimate?
5. How would you calculate the probability that fewer than 2 visitors are customers in the sample?

Tip: In binomial problems, the standard deviation depends not only on the probability but also heavily on the sample size. Increasing the sample size typically reduces uncertainty.