Question

A can of Pepsi is supposed to contain, on the average, 12 ounces of soda with a standard deviation of 0.3 ounces. Suspecting fraud, you take a random sample of 40 cans and measure the amount of Pepsi in each. Your measurements show that the 40 cans had a mean of11.9 ounces. What is the probability of a random sample of 40 cans having a mean of 11.9 ounces of soda or less?

Answer 1

@jim_thompson5910 this is my last question.

Answer 2

are you familiar with the xbar distribution?

Answer 3

It sounds familiar but I don't remember. How do I do that?

Answer 4

have you learned about the normal distribution at all?

Answer 5

Uhh, not that I really remember. I don't know.

Answer 6

ok one sec

Answer 7

heard of `Central Limit Theorem` ?

Answer 8

Nope. What's that?

Answer 9

you need to use CLT to get the normal distribution parameters,
and find the p-value for 11.9

Answer 10

First list down the given info :
\(\mu = 12\)
\(\sigma = 0.3\)
\(n=40\)

Answer 11

start by finding the standard error :

\(SE = \dfrac{\sigma}{\sqrt{n}} = ?\)

Answer 12

0.3/(srt40) = 0.04743416491
is that right?

Answer 13

thats right !

0.3/sqrt(40) is ur new standard deviation

Answer 14

CLT tells u that the new mean is same as population mean = 12

Answer 15

next find the zscore for 11.9

Answer 16

heard of `zscore` before right ?

Answer 17

How do I get zscore?

Answer 18

\(Zscore = \dfrac{\text{observation - mean}}{\text{standard deviation}} = \dfrac{11.9 - 12}{\frac{0.3}{\sqrt{40}}}\)

Answer 19

Once you have the Zscore,
lookup in Ztable for the corresponding probability

Answer 20

http://www.math.upenn.edu/~chhays/zscoretable.pdf

Answer 21

or u may use wolfram as well :

http://www.wolframalpha.com/input/?i=P%28X+%3C%3D++%5Cdfrac%7B11.9+-+12%7D%7B%5Cfrac%7B0.3%7D%7B%5Csqrt%7B40%7D%7D%7D%29

Answer 22

I got -2.108185106

Answer 23

thats right !
-2.108185106 is ur Zscore

Answer 24

to get the probability for that Zscore, u need to lookup in Ztable

Answer 25

http://www.math.upenn.edu/~chhays/zscoretable.pdf

Answer 26

know how to interpret Ztable ?

Answer 27

I don't understand how the ztable works...

Answer 28

its easy,
u familiar wid microsoft excel ?

Answer 29

a little

Answer 30

a little is enough :)

Answer 31

alright :) so how does it relate to the ztable?

Answer 32

-2.108185106

round it to two decimal places : -2.11

Answer 33

open this table :
http://www.math.upenn.edu/~chhays/zscoretable.pdf

Answer 34

0.0174?

Answer 35

You got it !

Answer 36

Really? Yay!

Answer 37

just fix ur one finger at ROW starting wid `-2.1`
and another finger at column having `0.01`

the intersection is ur probability : `0.0174`

Answer 38

yup ! thats all to Ztables :)

Answer 39

Ztable gives the probability (area under normal curve) for <= Zscore

(gives the area for left tail)

Answer 40

if above statement confuses, dont bother about it..

Answer 41

0.0174 is the `probability for having a sample of 40 cans whose mean is 11.9`

Answer 42

0.0174 is same as 1.74 % which is pretty less...

Answer 43

so, looks like there is indeed some fraud going on...

Answer 44

Really? So that's the answer to the question?

Answer 45

whats the question ?

Answer 46

this is ur question :

What is the probability of a random sample of 40 cans having a mean of 11.9 ounces of soda or less?

Answer 47

0.0174 is the probability
you're done.

Answer 48

if u prefer percentages, u can say the probability is 1.74 %

either way is fine

Answer 49

Really? Thank you! I really appreciate your help :)

Answer 50

np :) usually these questions are closely related to hypothesis testing... where they ask u figure out whether the fraud is going on or not...

Answer 51

for hypothesis testing - u need to work it the same way,
after finding the probability u need to make a decision also.

Answer 52

for example, here since the probability is very less, u might make a decision in favor of the claim that the fraud exists.