Question

1. Jamie believes that people who watch the news each night do better on a history test (out of 15 points) than people who do not. To test this theory, Jamie asks 7 people to watch the news for an hour each night before bed for one month, and then administers a historry test to these participants (in 2012). The following year, Jamie asks these same 7 people to avoid watching the news for one month, and then administers the history test one more time (in 2013). The data follows. Is Jamie right? (12 points) 2012 2013 9 4 12 6 10 7 10 14 6 10

Answer 1

$The\, \, \, result\, \, of\, \, a\, \, test\, \, of\, \, 7\, \, people\, \, who\, \, watch\, \, the\, \, news\, for\, \, an\, \, hour\, \, each\, \, night$

  $before\, \, bed\, \, for\, \, one\, \, month\, in\, \, 2012$

$9,\, \, 12,\, \, 6,\, \, 8,\, \, 9,\, \, 14\,, \, 10$

$The\, \, \, result\, \, of\, \, a\, \, test\, \, of\, \, 7\, \, people\, \, who\, \,avoid\, \, watching\, \, the\, \, news\, for$

  $one\, \, month\, in\, \, 2013$

$4,\, \, 5,\, \, 10,\, \, 7,\, \, 10,\, \, 6,\, \, 9$

$We\, \, first\, \, calculate\, \, difference\, \, between\, \, marks\, \, in\, \, test=x\, \, and\, \, from$

$these\, \, we\, \, calculate\, \, \bar{x}\, \, and\, \, s^{2}$

Tota 5 7 1 1 - 4 -718 3 -4 di ;-5 2 (-5) 2 153 / 6 8 16 4 3 6 gs

$\bar{x}=a+\frac{\sum d_{i}}{n}=5+\frac{\left ( -18 \right )}{7}=2.43$

$\sum \left ( x_{i-\bar{x}} \right )^{2}= \sum d_{i}^{2}-\frac{\left ( \sum d_{i} \right )^{2}}{n}=153-\frac{\left ( 18 \right )^{2}}{7}=106.71$

  $s^{2}=\frac{\sum \left ( x_{i} -\bar{x}\right )2}{n}=\frac{106.71}{7}=15.24$

  $s=\sqrt{15.24}=3.9$

$\left ( 1 \right )\, Null\, \, hypothesis\, \, H_{o}:\, \mu =0$

$Alternative\, \, hypothesis\, \, H_{a}:\, \mu > 0$

$\left ( 2 \right )\, Calculation\, \, of\, \, test\, \, statistic:\, since\, \,the\, \, sample\, \, size\, \, is\, \, small,\, we$

$use\, \, students\, t-distribution$

$\, t=\frac{\bar{x}-\mu }{\frac{s}{\sqrt{n-1}}}= \frac{2.43-0}{\frac{3.9}{\sqrt{7-1}}}=1.526$

$\left ( 3 \right )\, Level\, \, of\, \, confidence:\alpha =0.05$

$\left ( 4 \right )\, Critical\, \, value\, \, of\, \, t_{\alpha }\, \, at\, \, 5\, \, percent\, \, level\, \, of\, \, significance\, \, for\, \, degree$

  $of\, \, freedom\, \, v=7-1=6\, \, is\, \, 1.943\, \, for\, \, one\, \, tailed\, \, test$

$\left ( 5 \right )\, Decision:\, The\, \, calculated\, \, value\, \, of\, \, \left | t \right |=1.526\, \, is\, \, less\, \, than\, \, the$

$the\, \, critical\, \, value\, \, t_{\alpha }=1.943,\, the\, \, null\, \, hypothesis\, \, is\, \, accepted$

$\therefore \, Jamie believes\, \, that\, \, people\, \, who\, \, watch\, \, the\, \, news\, \, each\, \, night\, \, do\, \, better$

$on\, \, a\, \, history\, \, test\, \, than\, \, people\, \, who\, \, do\, \, not.\, \, The\, \, hypothesis\, shows\, \, that$

$Jamie\, \, is\, \, right$