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, 1 month ago

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Bill Foster commented 6 years, 1 month ago

Numbas Error: Error computing referenced variable popstdev

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Bill Foster 6 years, 1 month ago

Gave some feedback: Doesn't work

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Newcastle University Mathematics and Statistics 9 years, 2 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	45
mm	list	[ 0, 0, 1, 0 ]
thismany1	integer	30
thismany3	integer	100
units	string	pounds
popmean	number	1.27
message	string	We take a sample of size 100 a
thismuch	number	0.02
prob2	number	0.820
prob3	number	0.954
prob1	number	0.727

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random('petrol stations')

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Description

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

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Generated value: `string`

petrol stations

→ Used by:

"Part a)" - prompt
"Part b)" - prompt
"Part c)" - prompt

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, 1 month ago

Bill Foster 6 years, 1 month ago

Newcastle University Mathematics and Statistics 9 years, 2 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, 1 month ago

Bill Foster 6 years, 1 month ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: string

Parts

Number entry

Advanced settings

Generated value: `string`