An important component in the evaluating confidence interval

An important component in the evaluating confidence interval and performing hypothesis test is the critical value, and the critical value depends on what distribution we should apply. Please fill in the corresponding distribution (normal distribution or t distribution) in the following table.

Parameter

Requirements

Distribution for the Critical Value

One population Proportion p

(1) simple random sample
(2) conditions for the binomial distribution are satisfied
(3) np >= 5 and nq >= 5

One population Mean ?

(1) simple random sample
(2) the population is normally distributed or n > 30
(3) population standard deviation ? is unknown

One population Mean ?

(1) simple random sample
(2) the population is normally distributed
(3) population standard deviation ? is known

Solution

1. Normal because when np >= 5 and nq >= 5, we can approximate binomial distribution with normal distribution.

2. t distribution because population standard deviation is unknown.

3. Normal because population standard deviation is known.