Conditional probability in normal distribution,

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Question statement

Suppose that the height in {units1}, of $\\var{age}$ year old {mw} is a normal distribution with mean $\var{m}$ and standard deviation $\var{s}$.

Give any introductory information the student needs.

			Name	Type	Generated Value

units1	string	inches
lower	number	62
z	number	-0.5
age	integer	38
heightw	number	64.5
m	number	64.5
lower1	number	62.5
prop	number	30.9
heightm	number	72
s	integer	5
t	integer	1
w1	number	0.6554217416
prop1	number	58.3
mw	string	women
upper1	number	66
z1	number	-0.4
z2	number	0.3
w2	number	0.3820885778

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Name

Data type

Value

xxxxxxxxxx
 
'inches'

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Description

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Generated value: `string`

inches

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1. Gap Number entry
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Part b) Gap-fill
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What proportion of such {mw} are less than $\var{lower}$ {units1} tall?

Proportion =?Unnamed gap% (to 1 decimal place)..

Advice

Converting to $\operatorname{N}(0,1)$

\[\begin{eqnarray*}P(X \lt \var{lower})&=&P\left(Z \lt \frac{\var{lower}-\var{m}}{\var{s}}\right)\\&=&P(Z\lt \var{z})=1-P(Z \lt \var{-z})=1-\var{normalcdf(-z,0,1)}\\&=&\var{prop/100}\end{eqnarray*}\] to 3 decimal places.

Hence the proportion is $\var{prop}$% to 1 decimal place.

The proportion of such {mw} over $\var{lower1}$ {units1} tall that are more than $\var{upper1}$ {units1} tall is given by finding the conditional probability:

\[\begin{eqnarray*}P(X \gt \var{upper1}|X \gt\var{lower1})&=&\frac{P(X \gt \var{upper1} \text{ and } X \gt \var{lower1})}{P(X \gt \var{lower1})}\\&=&\frac{P(X \gt \var{upper1})}{P(X \gt \var{lower1})}\\&=&\frac{1-P(X \le \var{upper1})}{1-P(X \le \var{lower1})}\\&=&\frac{1-\Phi(\var{z2})}{1-\Phi(\var{z1})}\\&=&\frac{1-\Phi(\var{z2})}{\Phi(\var{-z1})}=\frac{\var{w2}}{\var{w1}}\\&=&\var{prop1/100}\end{eqnarray*}\]

to 3 decimal places.

hence the proportion is $\var{prop1}$% to 1 decimal place.

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Conditional probability in normal distribution,

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Conditional probability in normal distribution,

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Taxonomy: mathcentre

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Taxonomy: Context

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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