What is a sampling distribution? What purpose does it serve (i.e., how is it use

Question

What is a sampling distribution? What purpose does it serve (i.e., how is it used)? Derive the sampling distribution for the number of heads in 4 tosses of a biased coin with p (head) = 40. For each of the following, write the appropriate null and alternative hypotheses a) A researcher wants to determine if the mean weight of professional football players differs from 240 lbs. b) A researcher wants to determine if there is a positive correlation between level of education and income. c) A researcher wants to determine if there is a linear correlation between age and amount of alcohol consumption. d) A researcher wants to determine if the mean IQ test score for people who have graduated from college is greater than the national norm of 100. If X has a normal distribution with mu_x = 50 and sigma_x^2 = 25, if N = 16, what is the: a) mean of the sampling distribution of X? b) variance of the sampling distribution of X? c) standard error of the mean (the standard deviation of the sampling distribution of X)?

Explanation / Answer

SOLUTION:

1)Sampling distribution:The sampling distribution is a distribution of a sample statistic. When using a procedure that repeatedly samples from a population and each time computes the same sample statistic, the resulting distribution of sample statistics is a sampling distribution of that statistic.   

simply we can say that, sampling distribution is a graph of a statistic for your sample data.

for e.g. if we calculate sample mean then our sampling distribution would be the sampling distribution of the sample mean. likewise for median,variance,standard deviation,range etc. Every statistic has a sampling distribution

Now we will see how it is used:- we take example as follows:

suppose we have data as: 12,11,10,19,15,22 so a process takes a random sample without replacement of size N=3 from this population.  from this data are s1= {12,10,11},

S2={19,15,22} ......so, we have mean for the samples x1=11 , x2=18.67 ,.......If this process was continued indefinitely, then the infinite number of sample means would form a sampling distribution of the mean

instead of the mean, medians (Md) were computed for each sample. That is, within each sample the scores would be rank ordered and the middle score would be selected as the median then the sampling distribution of median we get from the data. so in this way we use sampling distribution.

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