Final answer:
To find the probability that the proportion of smokers in a sample of 766 females would be greater than 3%, we need to use the normal distribution. Calculate the mean and standard deviation of the sample proportion. Use the z-score formula to find the associated probability.
Step-by-step explanation:
To find the probability that the proportion of smokers in a sample of 766 females would be greater than 3%, we need to use the normal distribution. First, we need to calculate the mean and standard deviation of the sample proportion. The mean is equal to the population proportion, which is 4%, so the mean is 0.04. The standard deviation is calculated using the formula: sqrt((p * (1 - p)) / n), where p is the population proportion and n is the sample size. Substituting the values, we get sqrt((0.04 * (1 - 0.04)) / 766) = 0.0087. Next, we need to calculate the z-score using the formula: z = (x - mean) / standard deviation, where x is the desired proportion. Substituting the values, we get z = (0.03 - 0.04) / 0.0087 = -1.1494. Finally, we can use a standard normal distribution table or a calculator to find the probability associated with the z-score. For z = -1.1494, the probability is approximately 0.1251. Therefore, the probability that the proportion of smokers in a sample of 766 females would be greater than 3% is 0.1251.