Math 225N Week 5 Assignment_ {Graded A} | Central Limit Theorem for Means – Chamberlain College of Nursing

Week 5 Assignment: Central Limit Theorem for
Means


1. Question

A family of statisticians is trying to decide if they can afford for their child to play
youth baseball. The cost of joining a team is normally distributed with a mean of
$750 and a standard deviation of $185 . If a sample of 40 teams is selected at
random from the population, select the expected mean and standard deviation of
the sampling distribution below.

Correct answer:
σx¯=$29.25
μx¯=$750

The standard deviation of the sampling distribution

σx¯=σn−−√=$18540−−√=$29.25

When the distribution is normal the mean of the sampling distribution is equal to
the mean of the population μx¯=μ=$750 .


Question
A cupcake baker is planning a supplies order and needs to know how much flour he
needs. He knows that his recipes use an average of 100 grams of flour, normally
distributed, with a population standard deviation of 15 grams. If he is consulting a
sample size of 30 recipes, select the mean and standard deviation of the sampling
distribution to help him order his supplies from the options below.

σx¯=2.74 grams
μx¯=100 grams

,The standard deviation of the sampling distribution is

σx¯=σn−−√=1530−−√=2.74 grams
Likewise, when the distribution is normal the mean of the sampling distribution is
equal to the mean of the population μx¯=μ=100 grams.


Question
A head librarian for a large city is looking at the overdue fees per user system-wide
to determine if the library should extend its lending period. The average library
user has $19.67 in fees, with a standard deviation of $7.02 . The data is
normally distributed and a sample of 72 library users is selected at random from
the population. Select the expected mean and standard deviation of the sampling
distribution from the options below.

Correct answer:
σx¯=$0.83
μx¯=$19.67

The standard deviation of the sampling distribution is

σx¯=σn−−√=$7.0272−−√=$0.83

When the distribution is normal, the mean of the sampling distribution is equal to
the mean of the population μx¯=μ=$19.67 .




2. Question
A well known social media company is looking to expand their online presence by
creating another platform. They know that they current average 2,500,000
users each day, with a standard deviation of 625,000 users. If they randomly
sample 50 days to analyze the use of their existing technology, identify each of the
following, rounding to the nearest whole number if necessary:

, We are given population mean μ=2,500,000 and population standard
deviation σ=625,000 , and want to find the mean and standard error of the
sampling distribution, μx¯ and σx¯ for samples of size n=50 .

By the Central Limit Theorem, the means of the two distributions are the same:

μx¯=μ=2,500,000

To find the Standard Deviation of the sampling distribution, we divide the
population standard deviation by the square root of the sample size:

σx¯=σn−−√=625,00050−−√≈88,388



3. Question
A bank is reviewing its risk management policies with regards to mortgages. To
minimize the risk of lending, the bank wants to compare the typical mortgage owed
by their clients against other homebuyers. The average mortgage owed by
Americans is $306,500 , with a standard deviation of $24,500 . Suppose a
random sample of 150 Americans is selected.

Identify each of the following, rounding your answers to the nearest cent when
appropriate:

 1306500$306500$306500
 224500$24500$24500
 3150$150$150
 4306500$306500$306500
 $2000.42

We are given population mean μ=$306,500 and population standard deviation
σ=$24,500 , and want to find the mean and standard error of the sampling
distribution, μx¯ and σx¯ for samples of size n=150 .

By the Central Limit Theorem, the means of the two distributions are the same: