Central limit theorem

Question

The amount of coffee that people drink per day is normally distributed with a mean of 17 ounces and a standard deviation of 5.5 ounces. 34 randomly selected people are surveyed. Round all answers to 4 decimal places where possible.

C. What is the probability that one randomly selected person drinks between 16.7 and 17.4 ounces of coffee per day? 

D. For the 34 people find the probability that the average coffee consumption is between 16.7 and 17.4 ounces of coffee per day.

F. Find the IQR for the average of 34 coffee drinkers

Q1,Q3,IQR

Answer 1

X ~ N(17,30.25) where the normal distribution contains the parameters μ = 17 oz and σ² = 5.5² = 30.25 oz.

We need to use two different z-scores to find the probability for each one.

z₁ = (16.7 - 17) / 5.5 = -0.3/5.5 = -0.05

z₂ = (17.4 - 17) / 5.5 = 0.4/5.5 = 0.07

P(-0.05 < z < 0.07) = 0.5279 - 0.4801 = 0.0478

The probability that one randomly selected person drinks between 16.7 and 17.4 ounces of coffee per day is about 0.0478.

z₁ = (16.7 - 17) / (5.5 / √34) = -0.3 / 0.94324221828379860558258355372061 = -0.32

z₂ = (17.4 - 17) / (5.5 / √34) = 0.4 / 0.94324221828379860558258355372061 = 0.42

P(-0.32 < z < 0.42) = 0.6628 - 0.3745 = 0.2883

For the 34 people, the probability that the average coffee consumption is between 16.7 and 17.4 ounces of coffee per day is about 0.2883.

The IQR (Interquartile Range) for the average of 34 coffee drinkers can be calculated using the formula Q₃ - Q₁, where Q₁ is the first quartile and Q₃ is the third quartile.