Final answer:
To calculate the probability that the proportion of wrong numbers in a sample of 421 phone calls differs from the population proportion by greater than 3%, we can use the normal distribution.
Step-by-step explanation:
To calculate the probability that the proportion of wrong numbers in a sample of 421 phone calls differs from the population proportion by greater than 3%, we can use the normal distribution.
First, we need to find the standard deviation of the sample proportion, which is given by sqrt(p*(1-p)/n), where p is the population proportion and n is the sample size.
In this case, p = 0.08 and n = 421. Plugging these values into the formula gives us a standard deviation of 0.0146.
Next, we calculate the z-score for a difference of 3%, which is (0.03 - 0) / 0.0146 = 2.054.
We want to find the probability that the proportion differs by more than 3%, so we need to find the area under the curve to the left of -2.054 and to the right of 2.054.
Using a standard normal distribution table or a calculator, we find that the area to the left of -2.054 is 0.0202 and the area to the right of 2.054 is also 0.0202.
Therefore, the probability that the proportion differs by more than 3% is 2 * 0.0202 = 0.0404, or 0.0404 rounded to four decimal places.