The distribution of heights of adult males has a mean of 69 inches and a standard deviation of 4 inches. A random sample of 36 adult males is selected. Find the probability that the average height will be more than 70 inches.
a.0.668
b.0.858
c.0.908
d.0.995
To find the probability that the average height of a random sample of 36 adult males will be more than 70 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, regardless of the shape of the population distribution. In this case, as we have a large enough sample size (n = 36), we can assume that the distribution of sample means will be approximately normal.
The mean of the sample means (also known as the expected value of the sample mean) is equal to the population mean. So in this case, the mean of the sample means is 69 inches.
The standard deviation of the sample means (also known as the standard error of the mean) is equal to the population standard deviation divided by the square root of the sample size. So in this case, the standard deviation of the sample means is 4 inches divided by the square root of 36, which is 4/√36 = 4/6 = 2/3 inches.
To find the probability that the average height will be more than 70 inches, we can standardize the sample mean using z-scores and then use the standard normal table to find the probability.
The z-score for a sample mean x̄ can be calculated using the formula:
z = (x̄ - μ) / σ
where x̄ is the sample mean, μ is the population mean, and σ is the standard deviation of the sample means.
In this case, we want to find the probability of having a sample mean greater than 70 inches. So we calculate the z-score as:
z = (70 - 69) / (2/3) = (70 - 69) / (2/3) = 3/2
Using the standard normal table, we can find the probability of having a z-score greater than 3/2. The standard normal table gives us the cumulative probability to the left of a given z-score. Since we want the probability to the right of the z-score (greater than), we subtract the cumulative probability from 1.
Looking up the z-score of 3/2 (1.5) in the standard normal table, we find that the cumulative probability to the left of 1.5 is 0.9332. Subtracting this from 1 gives us the probability to the right of 1.5, which is 0.0668.
So the probability that the average height will be more than 70 inches is approximately 0.0668.
Since this question asks for the probability rounded to three decimal places, the correct answer is (a) 0.668.
To find the probability that the average height will be more than 70 inches, we need to convert the sample mean to a z-score and use the standard normal distribution.
Step 1: Calculate the standard error of the sample mean. This is obtained by dividing the standard deviation by the square root of the sample size:
Standard Error (SE) = Standard Deviation / Square Root of Sample Size
SE = 4 / √36
SE = 4 / 6
SE = 2/3
Step 2: Calculate the z-score by subtracting the population mean from the sample mean and dividing by the standard error:
z-score = (Sample Mean - Population Mean) / Standard Error
z-score = (70 - 69) / (2/3)
z-score = 3/2
Step 3: Look up the z-score in the standard normal distribution table (or use a calculator) to find the corresponding probability.
Using the standard normal distribution table or calculator, we find that the probability of a z-score of 1.5 or greater is approximately 0.9332.
Step 4: Subtract this probability from 1 to find the probability that the average height will be more than 70 inches:
Probability = 1 - 0.9332
Probability ≈ 0.0668
Therefore, the probability that the average height will be more than 70 inches is approximately 0.0668.
The correct answer is a. 0.668.
Z = (Sample mean-Pop. mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.