The heights of men in the USA are normally distributed with a mean of 68 inches and a standard of 4 inches. What is the probability that the mean height of a random sample of 35 men is greater than 69) inches? (Round your answer(s) to 3 decimal places.)
The heights of men in the USA are normally distributed with a mean of 68 inches and a standard of 4 inches. What is the probability that the mean height of a random sample of 35 men is greater than 69) inches? (Round your answer(s) to 3 decimal places.)
P(x > 69) =
To find the probability that the mean height of a random sample of 35 men is greater than 69 inches, we can use the Central Limit Theorem and the properties of a normal distribution.
According to the Central Limit Theorem, when the sample size is large enough, the distribution of sample means will be approximately normal, regardless of the shape of the population distribution. In this case, we have a sample size of 35, which is considered large enough for the Central Limit Theorem to apply.
Using the properties of a normal distribution, we can find the probability by standardizing the sample mean and using the standard normal distribution.
First, we calculate the standard error of the mean (SE) using the formula:
SE = σ / √n,
where σ is the standard deviation of the population (4 inches) and n is the sample size (35 men).
SE = 4 / √35 = 0.677
Next, we can calculate the z-score using the formula:
z = (x - μ) / SE,
where x is the value we want to find the probability for (69 inches) and μ is the mean of the population (68 inches).
z = (69 - 68) / 0.677 = 1.48
Finally, we can look up the probability of z > 1.48 in the standard normal distribution table or use a calculator or statistical software that provides this information.
Using a standard normal distribution table or calculator, we find that the probability of z > 1.48 is approximately 0.069.
Therefore, the probability that the mean height of a random sample of 35 men is greater than 69 inches is approximately 0.069.
P(x > 69) = 0.069 (rounded to 3 decimal places)