Question

#1 Describe the sampling distribution of ModifyingAbove p with caret. Assume the size of the population is 20 comma 000. nequals800​, pequals0.1 Choose the phrase that best describes the shape of the sampling distribution of ModifyingAbove p with caret below. A. Not normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis greater than or equals 10. B. Not normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis less than 10. C. Approximately normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis less than 10. D. Approximately normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis greater than or equals 10. Determine the mean of the sampling distribution of ModifyingAbove p with caret. mu Subscript ModifyingAbove p with caret Baseline equals nothing ​(Round to one decimal place as​ needed.) Determine the standard deviation of the sampling distribution of ModifyingAbove p with caret. sigma Subscript ModifyingAbove p with caret Baseline equals nothing ​(Round to three decimal places as​ needed.)

Answer 1

The framing of the question is such that it requires lot of reading and re reading of it to get an understanding of what exactly does it ask.

According to the question, the population proportion estimate is required to be derived and analysed from the data of the sample which has been carved out from it.

the sample size is 800 with p (probability/proportion) being equal to 0.10 .

The size of the population is 20000.

Before analysing the problem ,there is a concept that we need to know.

As the number of trials in a binomial experiment increases, the probability distribution becomes bell-shaped. As a rule of thumb, if np(1-p)≥10, the distribution will be approximately bell-shaped.

Let us check.

800*0.10*(1-0.10)

=72 >10

Thus from the data given in the problem we can visualize that the sampling distribution of will be bell shapped.

Infact if we observe the formula, we can see that it depends on the value of n.As n increases ,the size of the distribution tends to better bell curve.

Now that we know that the distribution is bell shaped, let us now develop few more conceptual understanding based on the following observations.

We have tasted that np(1-p)>=10

now lets test if or not.,

We take n/N =800/20000

=0.04

Clearly .

Hence we have validated two conditions and now we can say that since

and n*p*(1-p)>=10 ,our sampling distribution has to be a normal distribution (Bell curve being its hallmark)

Hence option (D)

Now let us try to find the mean

Clearly it is given by

where pis the population proprtion/probability

hence mean =0.10

This also is in line with our beleive inferred from the shape of the sampling distribution which is normal.