Lab 6:Central Limit Theorem

Exercise 1

A bottling company uses a filling machine to fill plastic bottles with soda. The bottles are supposed to contain 300 mL of soda each. It can be shown that the contents vary according to a normal distribution with a mean of 303 mL and a SD of 3 mL.

1. What is the probability that an individual bottle contains less than 300 mL?

1. Now take a random sample of 10 bottles. What is the mean and SD we can expect from the 10 bottles' average?

1. What is the probability that the sample mean volume of the 10 bottles is less than 300 mL?

Exercise 2

The weights of eggs produced by a certain breed of hen are normally distributed when a mean of 65g and a SD of 5g. A packaging company for a certain grocery store chain will only accept eggs that weigh between 63g and 68g.

1. If we randomly select an egg from a mass shipment, what is the probability that the egg will be within the store's standards?

1. Now consider a carton of 12 eggs. What is the probability that the average weight of the eggs is within the store's standards? What about a carton of 24 eggs?

1. Explain how and why the probability is changing the way it is from 2a to the each part of 2b.

Exercise 3

The length of human pregnancies from conception to birth varies according to a normal distribution with a mean of 264 days and a SD of 16 days. Consider 15 healthy, pregnant, women that are representative from this distribution.

1. What are the mean and SD of the sample mean length of pregnancy of these 15 women?

1. If we want to predict, with 90% accuracy, the sample mean length of pregnancy for the 15 women, what range of days would we find?

1. What's the probability the sample mean length of pregnancy lasts less than 205 days?

1. Toxic waste is believed to have affected the health of residence of this area. Suppose we are getting a high reporting of pregnancies lasting around 205 days for this area. Use your result from item 3c to argue that the waste is indeed affecting pregnancy length in women of this area.

Exercise 4

The soft-drink cups at a certain fast-food restaurant have tickets attached to them. Customers peel off the tickets to see if they win a prize. The proportion of winning tickets is 25%. A total of 70 people purchased soft-drinks between noon and 1:00 pm on a certain day.

1. What is the probability that at least a third of the people playing win a prize?

1. Using the 2-SD rule, what is the "usual range" of winning percentages we can expect?

1. If our local store in this chain reported having a sample proportion of winners that was representative of the 85th percentile of all stores carrying these tickets, what percentage of winners did they observe in their store?