Question

The average height of students in a freshman class of a certain school has been 161.27 cm with a population standard deviation of 6.5 cm. Is there a reason to believe that there has been a change in the average height if a random sample of 45 students in the present freshman class has an average height of 150.6 cm? Use a 0.05 level of significance.

Answer 1

$H_0:\mu =161.27\\H_1:\mu \ne 161.27\\Z=\sqrt{n}\frac{\bar{x}-\mu}{\sigma}=\sqrt{45}\frac{150.6-161.27}{6.5}=-11.0118\sim N\left( 0,1 \right) \\P-value:\\P\left( \left| Z \right|\geqslant 11.0118 \right) =2\varPhi \left( -11.0118 \right) =2\cdot 1.676\cdot 10^{-28}=3.352\times 10^{-28}\\Since\,\,the\,\,P-value\,\,is\,\,less\,\,than\,\,the\,\,level\,\,of\,\,significance, \mu \ne 161.27$