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?

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.

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

localid="1650891685864" $$\begin{gathered} SE=s_x/\surd n \\ SE=45/\surd 30 \\ SE=8.22 \end{gathered}$$

Test Hypothesis:

Null hypothesis $H₀x=\mu ₀$

Alternative hypothesis $Hₐx<\mu ₀$

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) localid="1650892596487" $p-value=0.062$

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

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