Final answer:
a. The probability is approximately 0.1179 or 11.79%. b. The probability is approximately 0.1664 or 16.64%. c. The normal distribution can be used in part (b) due to the Central Limit Theorem.
Step-by-step explanation:
a. To find the probability that an individual distance is greater than 206.80 cm, we need to calculate the z-score first. The z-score formula is z = (x - mean) / standard deviation. Plugging in the given values, we have z = (206.80 - 197.5) / 7.8 = 1.19. Using a standard normal distribution table or a calculator, we can find the probability associated with a z-score of 1.19. The probability is approximately 0.1179 or 11.79%.
b. To find the probability that the mean for 25 randomly selected distances is greater than 196.00 cm, we need to calculate the z-score for the sample mean. The formula for the standard error of the mean is standard deviation / sqrt(sample size). Plugging in the values, we have standard error = 7.8 / sqrt(25) = 1.56. We then calculate the z-score using the formula z = (sample mean - population mean) / standard error. Plugging in the remaining values, we have z = (196.00 - 197.5) / 1.56 = -0.96. Using a standard normal distribution table or a calculator, we can find the probability associated with a z-score of -0.96. The probability is approximately 0.1664 or 16.64%.
c. The normal distribution can be used in part (b) even though the sample size does not exceed 30 because of the Central Limit Theorem. The Central Limit Theorem states that regardless of the shape of the population distribution, the sampling distribution of the sample mean will approach a normal distribution as the sample size increases. As long as the sample size is reasonably large (usually considered to be larger than 30), the normal distribution assumption is valid.