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Stat 1 Pal Worksheet 22: Sampling Distribution of the Sample Mean (Continued)
Name______________________________
In the previous worksheet you were introduced to the random variable X = the number of pets of the entire population of N students at a large university.
X p( x ) (or P( X = x )) 0 0. 1 0. 2 0.
a. Students at this university own ๐๐ (^) ๐๐ = 0.5 pets, and that the standard deviation of the number of pets they own is ๐๐๐๐ = 0.6708. The distribution of the number of pets own by these students is skewed right. b. The population ๐๐๏ฟฝ^ of (^) ๐๐๐ถ๐ถ 2 possible means resulting from averaging the number of pets own by n = 2 students at a time has mean ๐๐ (^) ๐๐๏ฟฝ = 0.5 and standard deviation ๐๐๐๐๏ฟฝ = ๐๐๐๐/โ 2 = 0.4739. c. The distribution of the population of sample means ๐๐๏ฟฝ^ is closer to a bell-shape in comparison to the distribution of X.
1. Repeat the work you did in the previous worksheet by using now samples of n = 3 students at a time: the number of all possible samples you could get with n = 3 students from the population of N students would be (^) ๐๐๐ถ๐ถ 3. Suppose the number of students in this University is N = 20,000 and obtain (^20) , 000 ๐ถ๐ถ 3. That is, the population of all possible samples of n = 3 students
that can be made with the population of N = 20,000 students is (^20) , 000 ๐ถ๐ถ 3.
2. Even though the population of possible samples of n = 3 students from this university is very large ( (^) ๐๐๐ถ๐ถ 3 ), there are only 27 possible pairs of answers you would get when asking any two students the number of pets they have (each of which would repeat itself several times over
the entire population of (^) ๐๐๐ถ๐ถ 3 samples of 3 students). Compute the averages of the first 8 of these possible different answers would yield in the following table.
Answer of 1st^ Student ๐๐ 1
Answer of 2nd^ Student ๐๐ 2
Answer of 3rd Student ๐๐ 3
Mean Number of Pets ๐๐๏ฟฝ^ =
Etc. Etc. Etc. Etc.
3. In the above table that if you tried to list the population of (^) ๐๐๐ถ๐ถ 3 sample means ๐ฅ๐ฅฬ
resulting when asking each possible group of 3 students the number of pets they own, there are only a few possible different values (27) for these sample means. List all 4 of them:
4. Obtain the proportion of times that each of the 4 different ๐ฅ๐ฅฬ
โs you obtained above appear in the population of (^) ๐๐๐ถ๐ถ 3 ๐ฅ๐ฅฬ
โs:
Mean Number of Pets
๐๐๏ฟฝ^ =
p(๐ฅ๐ฅฬ
) (or P(๐๐๏ฟฝ^ = ๐ฅ๐ฅฬ
)
Etc.
5. The formulas mentioned in the previous worksheet, ๐๐ (^) ๐๐๏ฟฝ = ๐๐ (^) ๐๐ and ๐๐๐๐๏ฟฝ = ๐๐๐๐/โ๐๐ hold for samples of any size n. If you summarize the population of samples means ๐ฅ๐ฅฬ
โs you could get when averaging n values at a time from a population then: a. the average of the population of
๐๐^ ๐ถ๐ถ ๐๐sample means is always equal to the mean of the population from where you are taking your samples of size n, and b. the standard deviation of the population of (^) ๐๐๐ถ๐ถ ๐๐sample means is always smaller than the standard deviation of the values of the population from where you are taking your samples of size n by a factor of 1/โ๐๐.
Therefore, note that you donโt need to complete all the rows of the random variable ๐๐๏ฟฝ^ = ๐๐ 2 +๐๐ 2 +๐๐ 3 3 listed above to obtain its mean and standard deviation: ๐๐ (^) ๐๐๏ฟฝ = ๐๐๐๐๏ฟฝ =
6. In the previous worksheet that the shape of the distribution of ๐๐๏ฟฝ^ = ๐๐^2 +๐๐ 2 2 was closer to a
bell-shape than the distribution of the number of pets own by the students at this university X.
Based on this, how do you think the distribution of the random variable ๐๐๏ฟฝ^ = ๐๐^2 +๐๐ 32 +๐๐^3 looks
like in comparison to the distribution of these other two random variables?
7. Suppose a very large number of students who did not know that on average all student at the university own ๐๐ (^) ๐๐ = 0.5 pets, and each tried to estimate this value by averaging the number of pets own by a random sample of only three students. Will the proportion of students who get a sample mean ๐ฅ๐ฅฬ
that is only 0.6 units (either above or below) from the population mean ๐๐ (^) ๐๐= 0. be smaller, larger or equal to the proportion of students in the previous worksheet who are also only 0.6 units off from the mean ๐๐ (^) ๐๐= 0.5 but who used samples of n = 2 students? Explain.
