Sampling with and without replaement - binomial, hypergeometric, and Poisson approximation,

Error

There was an error loading the page.

Contributors

Newcastle University Mathematics and Statistics

History

○

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Created this.

There is only one version of this question that you have access to.

There are 4 other versions that do you not have access to.

Question statement

A {batch} of $\var{thismany}$ {items} has $\var{percent}$% {something}.

A sample of size $\var{s}$ is taken.

$X$ is the number of {something} in the sample.

Note that for the second part of this example we expect you to use R to calculate the probability.

Give any introductory information the student needs.

			Name	Type	Generated Value

bin	number	0.1841
tpapprox	number	0.1911531695
howmany	integer	3
thismany	integer	170
items	string	items
percent	integer	10
batch	string	batch
hyp	number	0.1766
p	rational	9/10
m	rational	3/2
papprox	number	0.1912
t	integer	0
tol	number	0.0001
s	integer	15
thyp	number	0.1766198192
tbin	number	0.1840610691
something	string	defectives

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
precround(tbin,4)

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: `number`

0.1841

← Depends on:

tbin

→ Used by:

Advice
"Part a)" → "Unnamed gap" - Minimum accepted value
"Part a)" → "Unnamed gap" - Maximum accepted value

Parts

Part a) Gap-fill
1. Gap Number entry
  Add
Part b) Gap-fill
1. Gap Number entry
  Add
Part c) Gap-fill
1. Gap Number entry
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Compute $P(X \ge \var{howmany})$ when sampling is with replacement.

$P(X \ge \var{howmany})=\;$Unnamed gap (to 4 decimal places).

Advice

For sampling with replacement we have $X \sim \operatorname{Bin}\left(\var{s},\var{percent/100}\right)$.

Hence:

$P(X \ge \var{howmany}) = 1-P(X \le \var{howmany-1})=1-\var{1-tbin}=\var{bin}$ to 4 decimal places.

For sampling without replacement we have $X \sim \operatorname{Hypergeometric}(N=\var{thismany},\;n=\var{s}, p=\var{percent/100})$.

Since $P(X \ge \var{howmany})=1-P(X \le \var{howmany-1})$ we calculate $P(X \le \var{howmany-1})$ using the hypergeometric probability distribution as calculated in R as we are counting the number of ways of selecting $\var{howmany-1}$ from $\var{round(thismany*percent/100)}$ defectives and the other $\var{s+1-howmany}$ in the sample from the $\var{round(thismany*(1-percent/100))}$ non-defectives.

The R expression we use for the probability is:

$\operatorname{phyper}(\var{howmany-1},\var{round(thismany*percent/100)},\var{round(thismany*(1-percent/100))},\var{s})=\var{1-thyp}$

Hence the answer for sampling without replacement is $1-\var{1-thyp}=\var{hyp}$ t0 4 decimal places.

We can approximate the random variable $X$ from part a) which follows a binomial distribution by the Poisson distribution $\operatorname{Poisson}\left(\mu=\var{s}\times \var{percent/100}\right)=\operatorname{Poisson}(\var{m})$.

We have $P(X \ge \var{howmany}) = 1-P(X \le \var{howmany-1})=1-\var{1-tpapprox}=\var{papprox}$ to 4 decimal places.

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet

This question is used in the following exams:

Loading...

Error

Sampling with and without replaement - binomial, hypergeometric, and Poisson approximation,

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: `number`

Parts

Gap-fill

Loading...

Error

Sampling with and without replaement - binomial, hypergeometric, and Poisson approximation,

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`