Q. 76 - The Practice of Statistics for AP

Question

Study more! A student group claims that first-year students at a university study $2.5$ hours per night during the school week. A skeptic suspects that they study less than that on average. He takes a random sample of $30$ first-year students and finds that $\overline{x} = 137$minutes and $s_x = 45$minutes. A graph of the data shows no outliers but some skewness. Carry out an appropriate significance test at the $5\%$ significance level. What conclusion do you draw?

Step-by-Step Solution

Answer

We are in the acceptance region we need not to reject $H₀$.

Step 1: Given information

A student group claims that first-year students at a university study $2.5$ hours per night during the school week. 

A skeptic suspects that they study less than that on average. 

He takes a random sample of $30$ first-year students and finds that $\overline{x} = 137$minutes and $s_x = 45$minutes. 

Step 2: Explanation

Normal distribution:

mean     $\mu ₀ = 150$

Sample:

Sample size    $n = 30$

Sample mean   $\overline{x} = 137$

Sample standard deviation  $s_x = 45$

The standard error of the sample mean  

$$\begin{gathered} SE = s_x /\surd n \\ SE = 45/\surd 30 \\ SE = 8.22 \end{gathered}$$

Test Hypothesis:

Null hypothesis                           $H₀ x = \mu ₀$ 

Alternative hypothesis                $Hₐ x < \mu ₀$

z(s)  test statistics is:

$$\begin{gathered} z\left(s\right) = ( x - \mu ₀ ) / s/\surd n \\ z\left(s\right) = - 13 /8.22 \\ z\left(s\right) = - 1.58 \end{gathered}$$

p-value  for that z(s)     $p-value = 0.062$

Then for $\alpha = 0.05 p-value > 0.05$

We are in the acceptance region we need not to reject$H₀.$