Final answer:
The probability that the mean height for the sample is greater than 64 inches is approximately 0.2863, or 28.63%.
Step-by-step explanation:
To find the probability that the mean height for the sample is greater than 64 inches, we need to use the Central Limit Theorem. The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as the sample size increases. We can use the standard normal distribution to calculate probabilities.
First, we need to calculate the standard error of the mean. The formula for the standard error of the mean is sigma / sqrt(n), where sigma is the population standard deviation and n is the sample size. In this case, sigma = 2.63 and n = 55. So the standard error is 2.63 / sqrt(55) = 0.3541.
Next, we calculate the z-score for the mean height of 64 inches. The formula for the z-score is (x - mu) / SE, where x is the sample mean, mu is the population mean, and SE is the standard error. In this case, x = 64, mu = 63.8, and SE = 0.3541. So the z-score is (64 - 63.8) / 0.3541 = 0.5653.
Finally, we can use the standard normal distribution to find the probability. We want to find the probability that the z-score is greater than 0.5653. Looking up this value in the standard normal distribution table, we find that the probability is approximately 0.2863, or 28.63%.