Is lighter better? A random sample of men's soccer shoes from an international catalog had the following weights (in ounces).

10.8 9.8 8.8 9.6 9.9 10 8.4 9.6 10 9.4 9.8 9.4 9.8

At $\alpha = 0.05$ can it be concluded that the average weight is less than 10 ounces?

#### Solution

Given that the sample size $n = 13$, sample mean $\overline{x}= 9.638$ and sample standard deviation $s = 0.585$.

**Hypothesis Testing Problem**

The hypothesis testing problem is

$H_0 : \mu = 10$ against $H_1 : \mu < 10$ ($\text{left-tailed}$)

**Test Statistic**

The test statistic is

` $$ \begin{aligned} t& =\frac{\overline{x} -\mu}{s/\sqrt{n}} \end{aligned} $$ `

which follows $t$ distribution with $n-1$ degrees of freedom.

**Significance Level**

The significance level is $\alpha = 0.05$.

**Critical Value**

As the alternative hypothesis is $\text{left-tailed}$, the critical value of $t$ for $12$ degrees of freedom $\text{is}$ $-1.782$.

The rejection region (i.e. critical region) is $\text{t < -1.782}$.

**Computation**

The test statistic under the null hypothesis is

` $$ \begin{aligned} t&=\frac{ \overline{x} -\mu_0}{s/\sqrt{n}}\\ &= \frac{9.638-10}{0.585/ \sqrt{13 }}\\ &= -2.227 \end{aligned} $$ `

**Decision**

*Traditional approach:*

The test statistic is $t =-2.227$ which falls $\text{inside}$ the critical region, we $\text{reject}$ the null hypothesis.

*$p$-value Approach*

This is a $\text{left-tailed}$ test, so the p-value is the

area to the left of the test statistic ($t=-2.227$) is p-value = $0.0229$.

The p-value is $0.0229$ which is $\text{less than}$ the significance level of $\alpha = 0.05$, we $\text{reject}$ the null hypothesis.

Thus the average weight is less than 10 ounces.