Question

Suppose that x has a distribution with μ = 8.5 and σ = 4. If a random sample of size n = 64 is chosen, find p(Xbar >10).

Answer 1

Given: μ = 8.5 and σ = 4 and Sample size (n) = 64 (large)

Standardizing : Assuming that the underlying distribution is normal (n= 64>30), (Central limit theorem:The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger, specifically for sample sizes over 30),

can be standardized into Z score as follows:

To find:

Substituting the given values,

  

Since, ,

The required probability can be obtained from Standard Normal Tables, looking for the area corresponding to Row 3.0 and column 0.00. Since the table provides the less than probabilities:

P(Z>3) = 1 - 0.99865

= 0.00135

Hence, = 0.00135