A telephone exchange operator assumes that $8 \%$ of the phone calls are wrong numbers.
If the operator is accurate, what is the probability that the proportion of wrong numbers in a sample of 598 phone calls would differ from the population proportion by greater than $3 \%$ ? Round your answer to four decimal places.
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Final Answer: The probability that the proportion of wrong numbers in a sample of 598 phone calls would differ from the population proportion by greater than 3% is approximately \(\boxed{0.0068}\).
Step 1 ：We are given a problem of sampling distribution. The population proportion (p) is 0.08 and we are asked to find the probability that the sample proportion (p_hat) differs from the population proportion by more than 0.03.
Step 2 ：We can use the Central Limit Theorem (CLT) to solve this problem. According to the CLT, if the sample size is large enough, the sampling distribution of the sample proportion is approximately normally distributed. The mean of this distribution is equal to the population proportion (p) and the standard deviation is \(\sqrt{\frac{p(1-p)}{n}}\), where n is the sample size.
Step 3 ：We can calculate the z-scores for the two boundaries (p - 0.03 and p + 0.03) and then use the standard normal distribution to find the probabilities. The z-score is calculated as \(\frac{p_hat - p}{std\_dev}\).
Step 4 ：Given p = 0.08, n = 598, and diff = 0.03, we calculate the standard deviation as 0.011094003924504582.
Step 5 ：We then calculate the z-scores for the two boundaries as -2.7041634565979917 and 2.7041634565979917.
Step 6 ：Using the standard normal distribution, we find the probabilities corresponding to these z-scores as 0.0034238297707716446 and 0.9965761702292284.
Step 7 ：The probability that the proportion of wrong numbers in a sample of 598 phone calls would differ from the population proportion by greater than 3% is the sum of these two probabilities, which is approximately 0.006847659541543205.
Step 8 ：Final Answer: The probability that the proportion of wrong numbers in a sample of 598 phone calls would differ from the population proportion by greater than 3% is approximately \(\boxed{0.0068}\).
