Question

1. For each of the following, explain if the Central Limit Theorem applies a) Estimating a right skewed distribution like income b) Estimating the mean of a right skewed distribution like income with a large sample size c)Finding the exact probability of getting a proportion of successes less than a value d) Creating an approximate confidence interval for a proportion assuming normality.

Answer 1

Solution:

central limit theorem ;

for sample size,n>30

sample follows normal distribution

with mean =x bar

and stddev=sample stddev/sqrt(n)

a) Estimating a right skewed distribution like income

we cannot use central limit theorem for estimating right skewed dsitribution

(b)

b) Estimating the mean of a right skewed distribution like income with a large sample size

we can use central limit theorem

for n>30

which is large sample

for right skewed distribution mean >median

and mean is at the center line

c)Finding the exact probability of getting a proportion of successes less than a value

we can use central limit theorem

if population follows normal distribution or sample size is large

we can find

P(p^<p)

p^=sample proportion

p-population proportion

z=p^-p/sqrt(p*(1-p)/n

P(Z<p) can be found from standard normal tables.

and probability is nothing but area under normal curve.

d) Creating an approximate confidence interval for a proportion assuming normality.

we can use central limit theorem

assuming nomrality

95% confidence interval for true population proportion can be given as

p^-z*sqrt(p^(1-p^)/n,p^+z*sqrt(p^(1-p^)/n

z alpha/2 for 95%=1.96