Final answer:
a. The probability that a randomly selected woman's height is less than 69.8 inches is approximately 0.992 or 99.2%. b. The probability that 70 randomly selected women will have a mean height of less than 69.8 inches is approximately 0.993 or 99.3%.
Step-by-step explanation:
a. To find the probability that a randomly selected woman's height is less than 69.8 inches, we need to calculate the z-score and then use the standard normal distribution table. The formula for calculating the z-score is: z = (x - mean) / standard deviation. Plugging in the given values, we get: z = (69.8 - 63.6) / 2.5 = 2.48. From the standard normal distribution table, we find that the probability corresponding to a z-score of 2.48 is approximately 0.992. Therefore, the probability that a randomly selected woman's height is less than 69.8 inches is approximately 0.992 or 99.2%.
b. To find the probability that 70 randomly selected women will have a mean height of less than 69.8 inches, we need to calculate the z-score for the sample mean. The formula for calculating the z-score for a sample mean is: z = (x - mean) / (standard deviation / sqrt(n)), where n is the sample size. Plugging in the given values, we get: z = (69.8 - 63.6) / (2.5 / sqrt(70)) = 2.488. From the standard normal distribution table, we find that the probability corresponding to a z-score of 2.488 is approximately 0.993. Therefore, the probability that 70 randomly selected women will have a mean height of less than 69.8 inches is approximately 0.993 or 99.3%.