Probability question stats

How many close friends do you have? Suppose that the number of close friends adults claim to have varies from person to person with mean = 5 and standard deviation = 1.8.

An opinion poll asks this question of an SRS of 900 adults.

We will see later that in this situation the sample mean response has approximately the Normal distribution with mean 5 and standard deviation 0.060.

What is , the probability that the sample result estimates the population truth = 5 to within 0.2?

Let Xbar represent the sample mean.
Xbar has approximately the normal distribution with mean 5 and standard deviation 0.060 .

P(Xbar is within 0.2 of 5)
= P(4.8 <= Xbar <= 5.2)
= P((4.8 - 5)/0.06 <= Z <= (5.2 - 5)/0.06)
= P(-3.33 <= Z <= 3.33)
= P(Z <= 3.33) - P(Z < -3.33)
= P(Z <= 3.33) - P(Z > 3.33) by symmetry of the normal curve
= P(Z <= 3.33) - [1 - P(Z <= 3.33)]
= 2P(Z <= 3.33) - 1
= 2(0.9996) - 1 from the normal table
= 0.9992 .

Lord bless you today!