assume that the heights of men are normally distributed with a mean of 68.4 inches and a standard deviation of 2.8 inches.if 64 men are randomly selected, find the probability that they have a mean height greater than 69.4 inches
To find the probability that 64 randomly selected men have a mean height greater than 69.4 inches, we can use the Central Limit Theorem. The Central Limit Theorem states that for a large enough sample size, the distribution of sample means will approach a normal distribution, regardless of the shape of the original population distribution.
Step 1: Calculate the standard deviation of the sample mean
The standard deviation of the sample mean, also known as the standard error, can be calculated using the formula: standard error = standard deviation / square root of sample size.
In this case, the standard deviation of the population (given) is 2.8 inches, and the sample size is 64. So, the standard error is 2.8 / √64 = 2.8 / 8 = 0.35 inches.
Step 2: Convert the given mean height to a z-score
To compare the sample mean height to the distribution of sample means, we need to convert it to a z-score. The formula for calculating a z-score is: z = (x - mean) / standard deviation.
In this case, the mean is 69.4 inches and the standard deviation is 0.35 inches (as calculated in Step 1).
So, the z-score is: z = (69.4 - 68.4) / 0.35 = 2.86.
Step 3: Find the probability using the z-table
Using a standard normal distribution (z-table), we can find the probability corresponding to a z-score of 2.86.
Find the value closest to 2.86 in the z-table, which is 2.9. The corresponding probability is 0.9981.
Since we are interested in the probability of having a mean height greater than 69.4 inches, we need to subtract this probability from 1.
So, the final probability is: 1 - 0.9981 = 0.0019.
Therefore, the probability that 64 randomly selected men have a mean height greater than 69.4 inches is approximately 0.0019.
To solve this problem, we can use the Central Limit Theorem. According to the theorem, the sample mean of a sufficiently large sample size from any population with a finite mean and a finite standard deviation will be approximately normally distributed.
In this case, the population mean is 68.4 inches, and the population standard deviation is 2.8 inches. Since the sample size is greater than 30 (64 men), we can assume that the sample mean will follow a normal distribution.
To find the probability that the sample mean is greater than 69.4 inches, we need to calculate the z-score and look it up in the standard normal distribution table.
First, calculate the standard error of the mean (SEM) using the formula:
SEM = σ / sqrt(n)
where σ is the population standard deviation and n is the sample size.
SEM = 2.8 / sqrt(64) = 2.8 / 8 = 0.35 inches
Next, calculate the z-score using the formula:
z-score = (sample mean - population mean) / SEM
z-score = (69.4 - 68.4) / 0.35 = 1 / 0.35 ≈ 2.857
Finally, using the z-score, we can find the probability using the standard normal distribution table or calculator. The probability can be calculated as the area under the curve to the right of the z-score.
Looking up the z-score of 2.857 in the standard normal distribution table, the probability is approximately 0.0022.
Therefore, the probability that the mean height of 64 randomly selected men is greater than 69.4 inches is approximately 0.0022, or 0.22%.