# 7.3 Using the central limit theorem  (Page 2/8)

 Page 2 / 8

Find the 90th percentile for the total of 75 stress scores. Draw a graph.

Let $k$ = the 90th percentile.

Find $k$ where $(P\left(\mathrm{\Sigma x}, k\right))=0.90$ .

$k=237.8$

The 90th percentile for the sum of 75 scores is about 237.8. This tells us that 90% of all the sums of 75 scores are no more than 237.8 and 10% are no less than 237.8.

invNorm $\left(.90,75\cdot 3,\sqrt{75}\cdot 1.15\right)=237.8$

Suppose that a market research analyst for a cell phone company conducts a study of their customers who exceed the time allowance included on their basic cell phone contract; the analyst finds that for those people who exceed the time included in their basic contract, the excess time used follows an exponential distribution with a mean of 22 minutes.

Consider a random sample of 80 customers who exceed the time allowance included in their basic cell phone contract.

Let $X$ = the excess time used by one INDIVIDUAL cell phone customer who exceeds his contracted time allowance.

$X$ ~ $\mathrm{Exp}\left(\frac{1}{22}\right)$ From Chapter 5, we know that $\mu =22$ and $\sigma =22$ .

Let $\overline{X}$ = the mean excess time used by a sample of $n=80$ customers who exceed their contracted time allowance.

$\overline{X}$ ~ $N\left(22,\frac{22}{\sqrt{80}}\right)$ by the CLT for Sample Means

## Using the clt to find probability:

• Find the probability that the mean excess time used by the 80 customers in the sample is longer than 20 minutes. This is asking us to find $(P\left(\overline{x}, 20\right))\phantom{\rule{20pt}{0ex}}$ Draw the graph.
• Suppose that one customer who exceeds the time limit for his cell phone contract is randomly selected. Find the probability that this individual customer's excess time is longer than 20 minutes. This is asking us to find $(P\left(x, \mathrm{20}\right))$
• Explain why the probabilities in (a) and (b) are different.

## Part a.

Find: $(P\left(\overline{x}, 20\right))$

$(P\left(\overline{x}, 20\right))=0.7919$ using normalcdf $\left(20,\mathrm{1E99},22,\frac{22}{\sqrt{80}}\right)$

The probability is 0.7919 that the mean excess time used is morethan 20 minutes, for a sample of 80 customers who exceed their contracted time allowance.

$\mathrm{1E99}={10}^{99}\text{and}\mathrm{-1E99}=-{10}^{99}$ . Press the EE key for E. Or just use 10^99 instead of 1E99.

## Part b.

Find P(x>20) . Remember to use the exponential distribution for an individual: X~Exp(1/22) .

P(X>20) = e^(–(1/22)*20) or e^(–.04545*20) = 0.4029

## Part c. explain why the probabilities in (a) and (b) are different.

• $(P\left(x, 20\right))=0.4029$ but $(P\left(\overline{x}, 20\right))=0.7919$
• The probabilities are not equal because we use different distributions to calculate the probability for individuals and for means.
• When asked to find the probability of an individual value, use the stated distribution of its random variable; do not use the CLT. Use the CLT with the normal distribution when you are being asked to find the probability for an mean.

## Using the clt to find percentiles:

Find the 95th percentile for the sample mean excess time for samples of 80 customers who exceed their basic contract time allowances. Draw a graph.

Let $k$ = the 95th percentile. Find $k$ where $(P\left(\overline{x}, k\right))=0.95$

$k=26.0$ using invNorm $\left(.95,22,\frac{22}{\sqrt{80}}\right)=26.0$

The 95th percentile for the sample mean excess time used is about 26.0 minutes for random samples of 80 customers who exceed their contractual allowed time.

95% of such samples would have means under 26 minutes; only 5% of such samples would have means above 26 minutes.

#### Questions & Answers

If potatoes cost Jane $1 per kilogram and she has$5 that could possibly spend on potatoes or other items. If she feels that the first kilogram of potatoes is worth $1.50, the second kilogram is worth$1.14, the third is worth $1.05 and subsequent kilograms are worth$0.30, how many kilograms of potatoes will she purchase? What if she only had $2 to spend? Susan Reply cause of poverty in urban DAVY Reply QI: (A) Asume the following cost data are for a purely competitive producer: At a product price Of$56. will this firm produce in the short run? Why Why not? If it is preferable to produce, what will be the profit-maximizing Or loss-minimizing Output? Explain. What economic profit or loss will the
Falak Reply
what is money
DAVY Reply
what is economic
Stephen Reply
economics is the study of ways in which people use resources to satisfy their wants
Falak
what is Price mechanism
Dhany Reply
introduction to economics
Uday Reply
welfare definition of economics
Uday
examine the wealth and welfare definitions of economics
Uday
read book by ml jhingan
Anand
What do we mean by Asian tigers
Aeesha Reply
Dm me I will tell u
Shailendra
Hi
Aeesha
hi
Pixel
What is Average revenue
KEMZO
How are u doing
KEMZO
it is so fantastic
metasebia
uday
Uday
it is a group of 4 countries named Singapore, South Korea, Taiwan and Hong Kong because their economies are growing very faster
Anand
fyn
EDWARD
Please, average revenue is an amount of money you gained after deducted your total expenditure from your total income.
EDWARD
what's a demand
Edward Reply
it is the quantity of commodities that consumers are willing and able to purchase at particular prices and at a given time
Munanag
quantity of commodities dgat consumers are willing to pat at particular price
Omed
demand depends upon 2 things 1wish to buy 2 have purchasing power of that deserving commodity except any from both can't be said demand.
Bashir
Demand is a various quantity of a commodities that a consumer is willing and able to buy at a particular price within a given period of time. All other things been equal.
Vedzi
State the law of demand
Vedzi
The desire to get something is called demand.
Mahabuba
what is the use of something should pay for its opportunity foregone to indicate?
Random Reply
Why in monopoly does the firm maximize profits when its marginal revenue equals marginal cost
astrid Reply
different between economic n history
Falma Reply
If it is known that the base change of RM45 million, the statutory proposal ratio of 7 per cent, and the public cash holding ratio of 5 per cent, what is the proposed ratio of bank surplus to generate a total deposit of RM 300 million?
Jeslyne Reply
In a single bank system, a bank can create a deposit when it receives a new deposit in cash. If a depositor puts a cash deposit of RM10,000 into the bank, assume the statutory reserve requirement is 7% and the bank adopts a surplus reserve of 8%. a. Calculate the amount of deposits made at the end o
Jeslyne
the part of marginal revenue product curve lies in the _ stage of production is called form demand curve for variable input.
Bashir Reply
The cost associated with the inputs owned by the farmer is termed as
Bashir
the cost associated with inputs owned by the farmer is termed as ____
Bashir
why do we study economic
Nwobodo Reply
we study economics to know how to manage our limited resources
Eben
တစ်ဦးကျဝင်​ငွေ
myo
we study economics the know how to use our resources and where to put it
Mamoud
what is end
Nwobodo
we study economics to make rational decision
Gloria
we study economics only to know how to effectively and efficiently allocate our limited resource in other to meet our unlimited wants
Kpegba
We study economics inorder for us to know the difference of the needs and wants and aslo how to use the limited resources that are available
Bongani
Difference between extinct and extici spicies
Amanpreet Reply
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.
Nerisha Reply

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics' conversation and receive update notifications?

 By Janet Forrester By Danielrosenberger By By Jonathan Long By Sean WiffleBoy By Brooke Delaney By OpenStax By By OpenStax By Maureen Miller