Question

Let ?̅be the sample mean of a continuous random variable ? for a sample with n ≥ 30 observations. Explain in your words how to compute the probability that the sample mean, ?̅, falls between two real values, ? and ?, with ? < ?, i.e.

Prob(? ≤ ?̅≤ ?)

Specify carefully all the assumptions you make and describe all the steps implemented.

Answer 1

Step 1: We need to know that wether the sample is normal or non-normal. Given that the sample size is n ≥ 30, we can assume that the sample comes from a normal population using the Central Limit Theorem.

Step 2: Once we know that the sample is normal, we need to know the population mean 'μ' (mu) and population standard deviation 'σ' (sigma).

Step 3: Next we need to calculate the z-score which is given by

z = x̅-μ / σ

where, z=standard score and x̅=observed value

Step 4: Now the probability can be calculated by normalising the values as follows:

Prob(? ≤ ?̅≤ ?) = Prob ( a-μ/σ ≤ x̅-μ/σ ≤ b-μ/σ ) = Prob ( za ≤ z ≤ zb ) = ϕ(zb) - ϕ(za)

where za=a-μ/σ and zb=b-μ/σ ; which are some numbers

Step 5: Now this probability can be calculated using the z score table.