Find the probability that a randomly selected sample of 30 men has a mean greater than 68 inches. The mean height of men is 69 inches and standard deviation of 2.8 inches
Z = (mean1 - mean2)/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 calculated.
To find the probability that a randomly selected sample of 30 men has a mean greater than 68 inches, we need to use the Central Limit Theorem and the properties of the normal distribution.
Step 1: Determine the distribution of the sample means
According to the Central Limit Theorem, the distribution of the sample means will be approximately normally distributed, regardless of the shape of the population distribution, as long as the sample size is sufficiently large (usually considered to be at least 30 in most cases). In this case, we have a sample size of 30 which satisfies this condition.
Step 2: Calculate the standard error of the mean
The standard error of the mean (SE) represents the standard deviation of the sampling distribution of the sample mean. It is calculated using the formula:
SE = standard deviation / square root of sample size
In this case, the standard deviation (σ) is given as 2.8 inches, and the sample size (n) is 30. Therefore, the standard error can be calculated as:
SE = 2.8 / √30 ≈ 0.511
Step 3: Standardize the sample mean
To find the probability, we need to standardize the sample mean so that we can refer to a standard normal distribution (mean of 0 and standard deviation of 1).
To standardize the sample mean, we calculate the z-score using the formula:
z = (sample mean - population mean) / standard error
In this case, the population mean (μ) is given as 69 inches, the sample mean is assumed to be 68 inches, and the standard error (SE) is 0.511. Therefore, the z-score can be calculated as:
z = (68 - 69) / 0.511 ≈ -1.961
Step 4: Find the probability using the standard normal distribution
Now that we have the z-score, we can use the standard normal distribution table or a statistical calculator to find the probability associated with the z-score.
The probability of a z-score less than -1.961 corresponds to the probability of obtaining a sample mean lower than 68 inches. Since we're interested in the probability of obtaining a sample mean greater than 68 inches, we can subtract this probability from 1.
You can use a standard normal distribution table or a statistical calculator to find that the probability of a z-score less than -1.961 is approximately 0.025. Therefore, the probability of a sample mean greater than 68 inches is approximately 1 - 0.025 = 0.975, or 97.5%.