Copyright [2020] [Alexander Craig Murph] Licensed under the
	Educational Community License, Version 2.0 (the "License"); you may
	not use this file except in compliance with the License. You may
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####### QUESTION AREA

<eqn>

########################
### Set General Params #
########################
# Country X and Y
$placeX = "Ireland";
$placeY = "US";
# Note that in Perl, 0=FALSE and 1=TRUE.
$reject = 0;
$alpha_level = 0.05;
$conf_level = (1 - $alpha_level)*100;
$alpha = $alpha_level/2;
if ($reject) {
  $reject_answer_false = "Fail to reject the null hypothesis";
  $reject_answer_true = "Reject the null hypothesis in favor of the mean.";
  $lower_fail = 1;
  $upper_fail = $alpha*1000 - 5;
} else {
  $reject_answer_true = "Fail to reject the null hypothesis";
  $reject_answer_false = "Reject the null hypothesis in favor of the mean.";
  $lower_fail = $alpha*1000 + 5;
  $upper_fail = $alpha*1000 + 250;
}


# Info for X: uniformly distributed
$lower_bound1 = 55;
$upper_bound1 = 90; 
$n1 = randnum(50, 75, 1);

# Info for Y: uniformly distributed
$lower_bound2 = 55;
$upper_bound2 = 90; 
$temp_n = $n1-1;
$n2 = randnum(30, $temp_n, 1);

$xincrement = ($upper_bound1 - $lower_bound1) / $n1;
$yincrement = ($upper_bound2 - $lower_bound2) / $n2;

$temp_n1 = $n1 - 1;
@index1 = (0..$temp_n1);
$temp_n2 = $n2 - 1;
@index2 = (0..$temp_n2);
my @includex; $includex[$n1] =' ';
my @includey; $includey[$n2] =' ';
my @datalist; $datalist[$n1] =' ';
my @datalisty; $datalisty[$n2] =' ';

for( $a = 0; $a < $n1; $a = $a + 1 ) {
   if ($index1[$a] <= $n1-1) {$includex[$a]=1};
}
for( $a = 0; $a < $n2; $a = $a + 1 ) {
   if ($index2[$a] <= $n2-1) {$includey[$a]=1};
}

my @x; $x[$n1] =0;
my @y; $y[$n2] =0;
my @d2; $d2[$n1] =0;
my @dy2; $dy2[$n2] =0;

for( $a =0; $a < $n1; $a = $a + 1 ) {
   $x[$a] = ((randnum(0,99,1)/100)*($upper_bound1 - $lower_bound1)/$n1)+$lower_bound1+($xincrement*$a);
   if ($includex[$a]==1) {$datalist[$a]=decform($x[$a],2)};
   if ($includex[$a]==1) {$d2[$a]=$datalist[$a]**2};
}

for( $a =0; $a < $n2; $a = $a + 1 ) {
   $y[$a]=((randnum(0,99,1)/100)*($upper_bound2 - $lower_bound2)/$n2)+$lower_bound2+($yincrement*$a);
   if ($includey[$a]==1) {$datalisty[$a]=decform($y[$a],2)};
   if ($includey[$a]==1) {$dy2[$a]=$datalisty[$a]**2};
}



# difference between means will fail a two sided p-value test 
$xbar = sum(@datalist)/$n1;
$ybar = sum(@datalisty)/$n2;
$ssx = sum(@d2) - ($n1*($xbar**2));
$ssy = sum(@dy2) - ($n2*($ybar**2));
$sdx = sqrt($ssx/($n1-1));
$sdy = sqrt($ssy/($n2-1));
$sd_mean = sqrt( (($sdx**2)/$n1) + (($sdy**2)/$n2) );
$diff_mean = $xbar - $ybar;


$lower_extreme_prob = randnum($lower_fail,$upper_fail,1)/1000;
$upper_tail = 1 - $lower_extreme_prob;
$mu_full = distr("cdf=normal, mu= 0 , s= 1 , prob= $upper_tail")*$sd_mean + $diff_mean;
$mu_full = decform($mu_full, 2) + 0.01;
$cdf_string = qq/cdf=normal, mu= $mu_full , s= $sd_mean , x= $diff_mean/;
$p_val = 2*(distr($cdf_string));


for( $a =0; $a < $n2; $a = $a + 1 ) {
   if ($includey[$a]==1) {$datalisty[$a]=$datalisty[$a] + $mu_full};
   if ($includey[$a]==1) {$dy2[$a]=$datalisty[$a]**2};
}

$xbar = sum(@datalist)/$n1;
$ybar = sum(@datalisty)/$n2;
$ssx = sum(@d2) - ($n1*($xbar**2));
$ssy = sum(@dy2) - ($n2*($ybar**2));
$sdx = sqrt($ssx/($n1-1));
$sdy = sqrt($ssy/($n2-1));
$sd_mean = sqrt( (($sdx**2)/$n1) + (($sdy**2)/$n2) );
$diff_mean = $xbar - $ybar;
$c = -distr("cdf=normal, mu=0, s=1, prob=$alpha");
$lower_ci = $diff_mean - $c*$sd_mean;
$upper_ci = $diff_mean + $c*$sd_mean;

$full_data = '';
for( $a =0; $a < $n1; $a = $a + 1 ) {
   if ($a <= $n2) {$full_data=$full_data . $datalist[$a] . ' ' . $datalisty[$a] . '<br>';};
   if ($a > $n2) {$full_data=$full_data . $datalist[$a] . '<br>';};
}

" "
</eqn>


Now you wish to know if the average height of people in the <eqn $placeY> is different than the average height of people in <eqn $placeX>.  You take a simple random sample of <eqn $n1> people from <eqn $placeX> (X) and <eqn $n2> people from the <eqn $placeY> (Y). <br><br>
a) What statistical test is appropriate for this research question? <_>
<br><br>
<SECTION>b) Let us choose a confidence level <latex>$\alpha = ${alpha_level}$ </latex> Choose the correct hypothesis: <_>
<br><br>
<SECTION>c) What conditions are required for inference? (Select all that apply)
<br><br>
<SECTION>d) Are these conditions all met?
<br><br>
<SECTION>e) What is our point estimate for difference in average height of people from these two areas? (for this entire problem, assume that you are performing inference on <latex> $\mu_X - \mu_Y$ </latex>) <_>
<br><br>
f) What is the sample standard deviation of the heights of people in <eqn $placeY>? <_><br>
What is the sample standard deviation of the heights of people in <eqn $placeX>? <_>
<br><br>
g) What is our approximation of the standard deviation of the difference between the average height of people in <eqn $placeY> and the average height of people in <eqn $placeX> (i.e. the standard error)? <_>
<br><br>
h) What is the p-value for this hypothesis test? <_>
<br><br>
<SECTION>i) What is your conclusion? <_>
<br><br>
<SECTION>j) Make a <EQN $conf_level>% confidence interval for the difference in the average heights between these two areas.<br> (<_>, <_>) <br>
<br>
<SECTION>k) Perform the hypothesis using your confidence interval.  Is your conclusion the same as the conclusion you made with your p-value? <_>
<br><br>
<u>Sample Data: (X, Y) in in:</u><br>
<eqn $full_data>
<br>
<br>






########### ANSWER AREA

Difference of the means for non-paired data.
Difference of the means for paired data.
Hypothesis test on heights of people in <EQN $placeX>.
Hypothesis test on heights of people in <EQN $placeY>.
<SECTION> <latex>$H_0: \mu_{X} = \mu_{Y}; H_a: \mu_{X} \neq \mu_{Y} $</latex>
<latex>$H_0: \mu_{X} \neq \mu_{Y}; H_a: \mu_{X} = \mu_{Y} $</latex>
<latex>$H_0: \mu_{X} > \mu_{Y}; H_a: \mu_{X} < \mu_{Y} $</latex>
<latex>$H_0: \mu_{X} < \mu_{Y}; H_a: \mu_{X} > \mu_{Y} $</latex>
<latex>$H_0: \mu_{X} \geq \mu_{Y}; H_a: \mu_{X} < \mu_{Y} $</latex>
<latex>$H_0: \mu_{X} \leq \mu_{Y}; H_a: \mu_{X} > \mu_{Y} $</latex>
<latex>$H_0: \mu_{X} > \mu_{Y}; H_a: \mu_{X} \leq \mu_{Y} $</latex>
<latex>$H_0: \mu_{X} < \mu_{Y}; H_a: \mu_{X} \geq \mu_{Y} $</latex>
<SECTION> Your sample of heights of people in EQN $placeY> is sufficiently large.
Your sample of heights of people in <EQN $placeX> is sufficiently large.
Your sample of heights of people in <EQN $placeY> is not strongly skewed.
Your sample of heights of people in <EQN $placeX> is not strongly skewed.
The observations in your sample of heights of people in <EQN $placeX> are independent.
The observations in your sample of heights of people in <EQN $placeY> are independent.
<SECTION> Yes.
No.
Not enough information.
<SECTION> <EQN $diff_mean> {tab} 0.001
<EQN $sdx> {tab} 0.001
<EQN $sdy> {tab} 0.001
<EQN $sd_mean> {tab} 0.001
<EQN $p_val> {tab} 0.001
<SECTION> <EQN $reject_answer_true>
<EQN $reject_answer_false>
Not enough information to make a conclusion.
<SECTION> <EQN $lower_ci> {tab} 0.001
<EQN $upper_ci> {tab} 0.001
<SECTION> Yes.
No.
Not enough information.