Is the Diet Practical? When 40 people used the Weight Watchers diet for one year, their mean weight loss was 3.0 lb and the standard deviation was 4.9 lb (based on data from "Comparison of the Atkins, Ornish, Weight Watchers, and Zone Diets for Weight Loss and Heart Disease Reduction," by Dansinger, et al., Journal of the American Medical Association, Vol. 293, No. 1). Use a 0.01 significance level to test the claim that the mean weight loss is greater than 0 lb. Based on these results, does the diet appear to be effective? Does the diet appear to have practical significance?
Solution
Given that the sample size $n = 40$, sample mean $\overline{x}= 3$ and sample standard deviation $s = 4.9$.
Hypothesis Testing Problem
The hypothesis testing problem is
$H_0 : \mu = 0$ against $H_1 : \mu > 0$ ($\text{right-tailed}$)
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 $\alpha = 0.01$.
Critical Value
As the alternative hypothesis is $\text{right-tailed}$, the critical value of $t$ for $39$ degrees of freedom $\text{is}$ $2.426$.
That is $P(t_{39}> t_{crit})=0.01 \Rightarrow t_{crit} =2.426$.
The rejection region (i.e. critical region) is $\text{t > 2.426}$.
Computation
The test statistic under the null hypothesis is
$$ \begin{aligned} t&=\frac{ \overline{x} -\mu_0}{s/\sqrt{n}}\\ &= \frac{3-0}{4.9/ \sqrt{40 }}\\ &= 3.872 \end{aligned} $$
Decision
Traditional approach:
The test statistic is $t =3.872$ which falls $\text{inside}$ the critical region, we $\text{reject}$ the null hypothesis.
$p$-value Approach
This is a $\text{right-tailed}$ test, so the p-value is the area to the left of the test statistic ($t=3.872$) is p-value = $P(t> 3.872)=0.0002$.
The p-value is $0.0002$ which is $\text{less than}$ the significance level of $\alpha = 0.01$, we $\text{reject}$ the null hypothesis.
We conclude that at $0.01$ level of significance the diet appears to be effective. That is the diet appears to have practical significance.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
