Consider the approximately normal population of heights of male college students with mean ì = 69 inches and standard deviation of ó = 4.6 inches. A random sample of 25 heights is obtained.
(c) Find the standard error of this sampling distribution. (Give your answer correct to two decimal places.)
Incorrect: Your answer is incorrect. .
(d) What is the probability that this sample mean will be between 40 and 55? (Give your answer correct to four decimal places.)
(e) What is the probability that the sample mean will have a value greater than 52? (Give your answer correct to four decimal places.)
(f) What is the probability that the sample mean will be within 6 units of the mean? (Give your answer correct to four decimal places.)
c) SEm = SD/√n
d) Z = (score-mean)/SEm
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to each Z score.
e) "Within 6 units of the mean" = 63-75. Use process above.
To find the standard error of the sampling distribution, you need to use the formula:
Standard Error = Standard Deviation / √(Sample Size)
In this case, the standard deviation (σ) is 4.6 inches and the sample size (n) is 25. So you can calculate the standard error as follows:
Standard Error = 4.6 / √(25)
Standard Error = 4.6 / 5
Standard Error ≈ 0.92
Therefore, the standard error of this sampling distribution is approximately 0.92 inches.
Now let's move on to the other questions:
(d) To find the probability that the sample mean will be between 40 and 55, you need to convert these values into z-scores and find the respective probabilities using the standard normal distribution table or calculator.
First, calculate the z-scores for 40 and 55 using the formula:
z = (x - μ) / σ
For 40:
z1 = (40 - 69) / 4.6
For 55:
z2 = (55 - 69) / 4.6
Once you have the z-scores, you can find the corresponding probabilities using the standard normal distribution table or calculator. The probability you are looking for is the difference between these two probabilities.
(e) To find the probability that the sample mean will have a value greater than 52, you need to calculate the z-score for 52 and then find the probability corresponding to that z-score using the standard normal distribution table or calculator.
(f) To find the probability that the sample mean will be within 6 units of the mean, you need to calculate the z-scores for 6 and -6 and then find the difference between the two probabilities using the standard normal distribution table or calculator.
Remember to always use standardized values (z-scores) when working with probabilities in a normal distribution.