Q1. A population has a mean of µ = 80 with σ = 20.
a. If a single score is randomly selected from this population, how much distance, on average, should you find between the score and the population mean?
20 because the standard deviation is 20
b. If a sample of n = 4 scores is randomly selected from this population, how much distance, on average, should you find between the sample mean and the population mean?
10 because 20/square root of 4 equals 10
c. What is the probability that sample mean will be less than 70 for a sample of 16 scores?
I don't know what formula to use for this one???
population has a mean of µ = 80 with σ = 20.
a. Would a score of X=70 be considered an extreme value (out in the tail) in this sample?
b. If the standard deviation were σ = 5, would a score of X=70 be considered an extreme value?
c. What is the probability that the sample mean will be less than 70 for a sample of 16 scores? Well, let's just say that the probability is as likely as finding a unicorn wearing a sombrero while riding a unicycle on a rainbow. In other words, it's pretty low! But, if you really want to calculate it, you can use the z-score formula and standard normal distribution tables. Good luck with that!
To calculate the probability that the sample mean will be less than 70 for a sample of 16 scores, you need to use the concept of the sampling distribution of the sample mean. The sampling distribution of the sample mean refers to the distribution of all possible sample means that could be obtained from the population.
In this case, you are given the population mean (µ = 80) and the population standard deviation (σ = 20). The formula for the standard deviation of the sampling distribution of the sample mean is σ/√n, where σ is the population standard deviation and n is the sample size.
In this case, the sample size is 16, so the standard deviation of the sampling distribution of the sample mean is 20/√16 = 5.
To find the probability that the sample mean will be less than 70, you can use the standard normal distribution (also known as the Z-distribution) since the sample mean follows a normal distribution due to the central limit theorem.
You need to calculate the Z-score for a sample mean of 70 using the formula Z = (X - µ) / (σ/√n). In this formula, X is the value you're interested in (70 in this case), µ is the population mean, σ is the population standard deviation, and n is the sample size.
Z = (70 - 80) / (5) = -2
Once you have the Z-score, you can use a Z-table or a statistical software to look up the probability associated with that Z-score. The probability will tell you the likelihood that a sample mean of 70 or less would be obtained from the population.
Please note that I'm simplifying the math involved, and in practice, a statistical software or Z-table is commonly used to find the probability accurately.
Z = (score-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/probability related to the Z score.