In this problem, we have a sample with n = 16, and the population has:
• mean value μ = 60,
• standard deviation σ = 20.
(a) We want to know the probability that the sample mean will be greater than 50. To compute this probability, we compute the z-score for x = 50:
[tex]z=\frac{x-\mu}{\sigma}=\frac{50-60}{20}=-0.5.[/tex]Using a table probability for the z-scores, we get:
[tex]P(X>50)=P(Z>-0.5)=0.6915.[/tex](b) We want to know the probability that the sample mean will be less than 50. To compute this probability, we compute the z-score for x = 56:
[tex]z=\frac{x-\mu}{\sigma}=\frac{56-60}{20}=-0.2.[/tex]Using a table probability for the z-scores, we get:
[tex]P(X<56)=P(Z<-0.2)=0.4207.[/tex](c) To determine the boundaries for the middle 80%, first, we find the z-scores values with a z-score table, we get:
Using these values, we compute the values of the boundaries:
Answer(a) Proportion = 0.6915
(b) Proportion = 0.4207
(c) Boundaries: x₁ = 34.364 and x₂ = 85.636
