Lab $6$ Sampling: Decision$-$Making Thresholds

A key focus of Chapter $10$ is how to make inferences about populations based on samples. The essential logic lies in comparing a single instance of a statistic, such as a sample mean, to a distribution of such values. The comparison can lead to one of two conclusions $$ the sample statistic is either extreme or not extreme. But what are the thresholds for making this kind of judgment call $($i$.$e$.,$ whether a value is extreme or not$)?$ This activity explores that question.

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The problem is this: You receive a sample containing the ages of $30$ students. You should be able to telwhether this sample is a group of undergraduates $($mean age $=20$ years$)$ or graduates $($mean age $=25$ years$).$ To answer this question, you must compare the mean of the sample you receive to a distribution of means from the population. The following fragment of R code begins the solution:

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#q$1$: this is to set seed. By doing so$,$ the initiation point is always the same, not random.

set.seed$(2)$

# assign the value $30$ to the variable "sampleSize"

sampleSize $<-30$

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#q$2$: randomly generate $20000$ numbers that obeyed normal distribution with mean $20$ and standard deviation $3.$

# Then assign those numbers to the vector "studentPop".

studentPop $<-$ rnorm$(20000,$mean$=20,$sd$=3)$

#investigate studentPop now. How many rows? What are the values look like? Are they close to the mean value $20?$

#q$3$: draw sampleSize $(30$ sample defined above$)$ from studentPop and assign the $30$ numbers to "undergrads".

undergrads $<-$ sample$($studentPop$,$size$=$sampleSize,replace$=$TRUE$)$

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#q$4$: create a sample of graduate students$--$randomly generate $30$ numbers $($use sampleSize$)$ that obeyed normal distribution with mean $25$ and sd $3.$ See the mean is $5$ years older than the undergraduate sample apparently. $$

# Assign the numbers to the vector "grads".

# Sample size is sampleSize $(==30),$ mean is $25,$ standard deviation is $3.$

grads $<-$ rnorm$($sampleSize$,$mean$=25,$sd$=3)$

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#q$5$: Randomly assign either the grads sample or the undergrads sample to testSample, depending on the value generated by runif$(1).$

# "runif$(1)"$ would generate a random number between $0$ and $1.$

# If the number is greater than $0\mathrm{.}5,$ assign grads sample to testSample. Otherwise, assign undergrads sample to testSample.

if $($runif$(1)>0\mathrm{.}5)\{$ testSample $<-$ grads $\}$ else $\{$ testSample $<-$ undergrads $\}$

mean$($testSample$)$

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#q$6$: calculate the mean of "testSample" What is the mean of testSample?

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After you run this code, the variable $$testSample$$ will contain either a sample of undergrads or a sample of grads. The line before last $$flips a coin$$ by generating one value from a uniform distribution $($by default the distribution covers $0$ to $1)$ and comparing it to $0\mathrm{.}5.$ The question you must answer with additional code is: Which is it$,$ grad or undergrad?

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#q$7$: Generate $100$ sample means from studentPop and assign it as mySample

mySample $<-$ replicate$(---$your code here$-----)$

#q$8$: Compare mean$($testSample$)$ to that list of sample means $($mySample$)$ and see where it falls. quantile$()$ function helps display data distribution. Produce quantiles on thresholds $2\mathrm{.}5\%$ and $97\mathrm{.}5\%.$

$---$your code here$-----$

#q$9$: if the sample mean is less than quantiles on thresholds $2\mathrm{.}5\%$ or greater than quantiles on thresholds $97\mathrm{.}5\%,$ then it can be definded as extreme. Otherwise it is not extreme.

if $($mean$($object name here$)<$ quantile$($object name here, probs$=0\mathrm{.}025)|$ mean$($object name here$)>$quantile$($object name here, probs$=0\mathrm{.}975))$

$\{$ print$("$Sample mean is extreme"$)\}$ else $\{$ print$("$Sample mean is not extreme"$)\}$

$-----$