Advertisers contract with Internet service providers and search engines to place ads on websites. They pay a fee based on the number of potential customers who click on their ad. Unfortunately, click fraud—the practice of someone clicking on an ad solely for the purpose of driving up advertising revenue—has become a problem. Forty percent of advertisers claim they have been a victim of click fraud (BusinessWeek, March 13, 2006). Suppose a simple random sample of 380 advertisers will be taken to learn more about how they are affected by this practice.

a. What is the probability that the sample proportion will be within .04 of the population proportion experiencing click fraud? .95
b. What is the probability that the sample proportion will be greater than .45?

To answer these questions, we need to use the properties of the sampling distribution of the sample proportion.

a. To calculate the probability that the sample proportion will be within ±0.04 of the population proportion experiencing click fraud, we need to find the standard deviation of the sampling distribution.

The formula for the standard deviation of the sampling distribution, also known as the standard error, is given by:

SE = sqrt((p * (1 - p)) / n)

Where:
- p is the population proportion experiencing click fraud (unknown).
- n is the sample size.

Since we do not know the population proportion p, we will use the worst-case scenario that maximizes the standard error. This is when p = 0.5 (which gives the maximum variability in the proportions).

Using the given sample size of n = 380, we can calculate the standard error:

SE = sqrt((0.5 * (1 - 0.5)) / 380) ≈ 0.025063

Next, we need to calculate the z-score corresponding to the desired confidence level of 95%. The z-score represents the number of standard deviations away from the mean proportion.

For a two-tailed test at the 95% confidence level, the critical z-value is 1.96 (obtained from the standard normal distribution table).

Now we can calculate the margin of error (E) by multiplying the standard error by the critical z-value:

E = 0.025063 * 1.96 ≈ 0.049110

Finally, the probability that the sample proportion will be inside ±0.04 of the population proportion can be calculated by subtracting the margin of error from 1 (as we want the probability of being within the stated range):

Probability = 1 - E ≈ 1 - 0.049110 ≈ 0.950890

Therefore, the probability that the sample proportion will be within ±0.04 of the population proportion experiencing click fraud is approximately 0.9509, or 95.09%.

b. To calculate the probability that the sample proportion will be greater than 0.45, we need to find the z-score corresponding to this proportion and calculate the probability to the right of that z-score.

First, we need to calculate the standard error using the same formula as in part a:

SE = sqrt((p * (1 - p)) / n)

Again, using p = 0.5 to maximize the standard error:

SE = sqrt((0.5 * (1 - 0.5)) / 380) ≈ 0.025063

Next, we calculate the z-score for the sample proportion of 0.45:

z = (0.45 - p) / SE

Since we are interested in the probability that the sample proportion is greater than 0.45, we need to find the area to the right of this z-score in the standard normal distribution.

Using a standard normal distribution table or calculator, we find that the z-score is approximately 1.79.

Finally, we can calculate the probability that the sample proportion will be greater than 0.45 by subtracting the cumulative probability up to that z-score from 1:

Probability = 1 - Probability(z < 1.79)

The exact probability depends on the specific values in the standard normal distribution table or calculator.