a. What do the Law of Large Numbers and Gosset’s Theorem tell us about why the v

Question

a. What do the Law of Large Numbers and Gosset’s Theorem tell us about why the variances and ranges of sample means of these two distribution differ so considerably.

b What is the advantage of drawing large samples when carrying out a sampling experiment?

ll ustribution your Calculate tion, we no. 1 onl response Why possible. the variance of the population, if c) Can you determine the general shape of thd tribution from the data you have been given Why or why not? d) W hat theorem have you been relying on in responding to questions about this distribution and how did you determine this? ts of sampling experiments fronm two tygoeical peraltan Table 8E. Population and sample size Mean of sample mean Variance of sample means Standard deviation o sample means Range 24.01 Distribution no. 1 N = 824 0.68 0.82 23.64-25.03 24.02 57.23 Distribution no. 2 N=25 7.57 16.50 -3132 Can you determine the general shape of the dis tribution from the data you have been given?. b) 2. Now let's turn to distribution no. 2. Why or why not? about this distribution? If the theorem is dif c) What is the population mean for this dist ferent from the one you used in Question1, a) What theorem will you draw on to learn more bution? How did you determine this value d) Calculate the variance of the population m? If the

Explanation / Answer

Part (a) : Law of Large numbers and Gosset's Theorem tells that - If the sample size is very large, we may have precise information about the value of the true population mean based on the sample mean. But, if our sample is small, there are two sources of uncertainty - (1) Due to the "Error of random sampling", sample mean deviates widely from the population mean. (2) Lack of knowledge about the true distribution of the parameter of interest. Therefore, the particular observation we have in this data is explained by Law of Large Numbers and Gosset Theorem in that "as we increase the sample size, we will see that the Variances in the distribution of sample mean reduces; and so does range of sample mean values." Hence, with sample size N = 824, we have lesser standard deviation and range for sample mean, than with sample size N = 25.

Part (b) : The advantage of drawing large samples when carrying out a sampling experiment is that we can have more precise information about the population mean and the particular estimate of the sample mean computed will be much more reliable (i.e. low variance).

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