Question

There are 4 conditions that must be true in order to use the Central Limit theorem. 1) We must have a simple random sample (SRS); 2) the sample size must be less than 10% of the population; 3) the observations must be independent; and 4) the sample size must be large enough so that both np > 10 and n(1 - p) >10, in which the true proportion (or probability) possessing the attribute of interest is p. Then the Central Limit Theorem predicts three things about the sampling distribution of the sample proportion :

1. Shape: The distribution is approximately normal.

2. Center: The mean will equal p.

3. Spread: The standard deviation will equal p(1-p)n

Note: This normal approximation becomes more and more accurate as the sample size increases and is generally considered to be valid as long as np > 10 and n(1 - p) >10.

1. If you continue to assume that the population proportion of coin flips is .05 what does the CLT predict for the shape of the sampling distribution and for the values of the mean and standard deviation of the sampling distribution of sample proportions when the sample consists of a given number of flips? Using the formulas above, complete the table below.

	n = 20	n = 60	n = 180
Shape
Center (mean)
Spread (SD)

Answer 1

Answer)

For n = 20

N*p = 20*0.5 = 10

N*(1-p) = 10

Since both the conditions are met

Shape = normal (symmetric)

Mean = n*p = 10

S.d = √{n*p*(1-p)} = 2.23606797749

For n = 60

N*p = 30

N*(1-p) = 30

Shape = normal

Mean = n*p = 30

S.d = √{n*p*(1-p)} = 3.87298334620

For n = 180

N*p = 90

N*(1-p) = 90

Shape = normal.

Mean = 90

S.d = 6.70820393249