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.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
