Question

A bottling company prints "300 ml" on its label. The quality control supervisor selects nineteen cans at random and checks them. She finds the sample mean volume to be x = 297.3 and 5 = 3.2. Do the data present sufficient evidence to indicate that the mean volume is less than that claimed on the label? (Use a = 0.05.) (a) State the null and alternative hypotheses. O Hou # 300 versus H: Il = 300 OH: < 300 versus H: U > 300 Ho: 4 = 300 versus H: 300 O Ho: 4 = 300 versus H: 4 < 300 OH:4< 300 versus H: 4 = 300 (b) State the test statistic. (Round your answer to three decimal places.) (c) State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three decimal places.) (d) State the conclusion. Hy is not rejected. There is sufficient evidence to conclude that the mean volume is less than that claimed on the label. O H, is rejected. There is insufficient evidence to conclude that the mean volume is less than that claimed on the label. OH, is not rejected. There is insufficient evidence to conclude that the mean volume is less than that claimed on the label. O Ho is rejected. There is sufficient evidence to conclude that the mean volume is less than that claimed on the label. You may need to use the appropriate appendix table or technology to answer this question.

Answer 1

$Given\, \, \mu =300,\, \, \bar{X}=297.3,\, \, standard\, \, deviation\, \, of\, \, sample\, \, s=3.2$

$\left ( a \right )\textbf{Null Hypothesis}\, \, H_{0}:\, \, \mu =300$

$\textbf{Alternative Hypothesis}\, \, H_{0}:\, \, \mu < 300$

$\left ( b \right )\textbf{Text statistic}$

$t=\frac{\bar{X}-\mu }{\frac{s}{\sqrt{n}}}=\frac{297.3-300}{\frac{3.2}{\sqrt{19}}}$

$=\textbf{-3.678}$

$\textbf{Level of significance}:\, \, \alpha =0.05$

$\textbf{Critical value}:\, \, The\, \, value\, \, \, of\, \, t_{\alpha }\, \, at\, \, \alpha =0.05\, \, level\, \, of\, \, significance\, \, for$

$\left ( c \right )\, \, State\, \, the\, \, rejection\, \, region$

  $\textbf{The rejection region is t}< \textbf{-2.101}\, \, for\, \, left\, \, tailed\, \, test$

$As\, \, t_{calculated}\left ( -3.678 \right )\, \, is\, \, less\, \, than\, \, t_{\alpha }\left ( -2.101 \right )$

$\left ( d \right )\, \, H_{o}\, \, \textbf{is rejected. There is sufficient evidence to conclude that}$

  $\textbf{the mean volume is less than that claimed on the label}$