Probability sample mean is within d of the population mean

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Bill Foster

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Doesn't work

6 years, 2 months ago

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Bill Foster commented 6 years, 2 months ago

Numbas Error: Error computing referenced variable popstdev

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Bill Foster 6 years, 2 months ago

Gave some feedback: Doesn't work

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Question statement

The population mean of the {this} of {that} is $\var{popmean}$ {units} with population standard deviation $\var{popstdev}$ {units}.

Find the following probabilities.

Input all probabilities to 3 decimal places.

Give any introductory information the student needs.

			Name	Type	Generated Value

these	string	petrol stations
generateprobs	list	List of 6 items
that	string	unleaded petrol in the UK per
this	string	price
popstdev	number	0.1
thismany2	integer	50
mm	list	[ 0, 0, 0, 1 ]
thismany1	integer	30
thismany3	integer	90
units	string	pounds
popmean	number	1.26
message	string	None of the probabilities is $
thismuch	number	0.02
prob2	number	0.843
prob3	number	0.942
prob1	number	0.727

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random('petrol stations')

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `string`

petrol stations

Advice

Let $M$ be the mean price of the sample

Converting to $N(0,1)$ we see that we require the probability that if $\displaystyle Z=\frac{|M-\var{popmean}|}{\frac{\var{popstdev}}{\sqrt{\var{thismany1}}}}$ then we require the probability that \[\frac{-\var{thismuch}}{\frac{\var{popstdev}}{\sqrt{\var{thismany1}}}} \lt Z \lt \frac{\var{thismuch}}{\frac{\var{popstdev}}{\sqrt{\var{thismany1}}}} \implies \var{-thismany1^0.5*thismuch/popstdev}\lt Z \lt \var{thismany1^0.5*thismuch/popstdev}\].

The probability that $Z \lt \var{-thismany1^0.5*thismuch/popstdev}$ is $p=\var{normalcdf(-thismany1^0.5*thismuch/popstdev,0,1)}$.

Hence the probability we want is $P_1=1-2p=\var{prob1}$ to 3 decimal places.

Similarly we have for sample size $\var{thismany2}$ we want the probability that:

\[\frac{-\var{thismuch}}{\frac{\var{popstdev}}{\sqrt{\var{thismany2}}}} \lt Z \lt \frac{\var{thismuch}}{\frac{\var{popstdev}}{\sqrt{\var{thismany2}}}} \implies \var{-thismany2^0.5*thismuch/popstdev}\lt Z \lt \var{thismany2^0.5*thismuch/popstdev}\].

The probability that $Z \lt \var{-thismany2^0.5*thismuch/popstdev}$ is $p=\var{normalcdf(-thismany2^0.5*thismuch/popstdev,0,1)}$.

Hence the probability we want is $P_2=1-2p=\var{prob2}$ to 3 decimal places.

For sample size $\var{thismany3}$ we want the probability that:

\[\frac{-\var{thismuch}}{\frac{\var{popstdev}}{\sqrt{\var{thismany3}}}} \lt Z \lt \frac{\var{thismuch}}{\frac{\var{popstdev}}{\sqrt{\var{thismany3}}}} \implies \var{-thismany3^0.5*thismuch/popstdev}\lt Z \lt \var{thismany3^0.5*thismuch/popstdev}\].

The probability that $Z \lt \var{-thismany3^0.5*thismuch/popstdev}$ is $p=\var{normalcdf(-thismany3^0.5*thismuch/popstdev,0,1)}$.

Hence the probability we want is $P_3=1-2p=\var{prob3}$ to 3 decimal places.

d) {message}

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Probability sample mean is within d of the population mean

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Bill Foster commented 6 years, 2 months ago

Bill Foster 6 years, 2 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: `string`

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Error

Probability sample mean is within d of the population mean

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster commented 6 years, 2 months ago

Bill Foster 6 years, 2 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: string

Parts

Number entry

Advanced settings

Generated value: `string`