A crossover study was conducted to investigate whether oat bran cereal helps to lower serum cholesterol levels in hypercholesterolemic males. Fourteen such individuals were randomly placed on a diet that included either oat bran or corn flakes; after two weeks, their low density lipoprotein (LDL) cholesterol levels were recorded. Each man was then switched to the alternative diet. After a second two week period, the LDL cholesterol level of each individual was again recorded. The data from this study are shown below.

Subject LDL (mmol/l) Corn Flakes LDL (mmol/l) Oat Bran
1 4.61 3.84
2 6.42 5.57
3 5.40 5.85
4 4.54 4.80
5 3.98 3.68
6 3.82 2.96
7 5.01 4.41
8 4.34 3.72
9 3.80 3.49
10 4.56 3.84
11 5.35 5.26
12 3.89 3.73
13 2.25 1.84
14 4.24 4.14

a) Are the two samples of data paired or independent?
b) What are the appropriate Null and alternative hypothesis for a two sided test?
c) Conduct the test at the 0.05 level of significance. What is the p-value?
d) What do you conclude?

Solution

a) The two samples of data are paired. One measurement while eating one cereal, another while eating the other cereal, both on the same person.

b) Given that the sample size $n = 14$. Let $d=x-y$.

The appropriate null and alternative hypothesis for two-sided test is
$H_0 : \mu_d = 0$ against $H_1 : \mu_d \neq 0$ ($\textit{two-tailed}$)

x y d d-dbar (d-dbar)^2
4.61 3.84 0.77 0.4071429 0.1657653
6.42 5.57 0.85 0.4871429 0.2373082
5.40 5.85 -0.45 -0.8128571 0.6607367
4.54 4.80 -0.26 -0.6228571 0.3879510
3.98 3.68 0.30 -0.0628571 0.0039510
3.82 2.96 0.86 0.4971429 0.2471510
5.01 4.41 0.60 0.2371429 0.0562367
4.34 3.72 0.62 0.2571429 0.0661224
3.80 3.49 0.31 -0.0528571 0.0027939
4.56 3.84 0.72 0.3571429 0.1275510
5.35 5.26 0.09 -0.2728571 0.0744510
3.89 3.73 0.16 -0.2028571 0.0411510
2.25 1.84 0.41 0.0471429 0.0022224
4.24 4.14 0.10 -0.2628571 0.0690939

$\overline{d}= 0.3629$ and $s_d = 0.406$.

c) Test Statistic

The test statistic is
$$ \begin{aligned} t=\frac{\overline{d} -\mu_d}{s_d/\sqrt{n}} \end{aligned} $$

Level of Significance

The significance level is $\alpha = 0.05$.

Computation

The test statistic for testing above hypothesis testing problem under the null hypothesis is
$$ \begin{aligned} t&=\frac{\overline{d} -\mu_d}{s_d/\sqrt{n}}\\ &= \frac{0.3629-0}{0.406/\sqrt{14}}\\ &= 3.3445 \end{aligned} $$

$p$-value approach:

The test is $\textit{two-tailed}$ test, so p-value is the area to the $\textit{extreme}$ of the test statistic ($t=3.3445$). That is p-value = $2*P(t\geq 3.3445 ) = 0.0053$.

The p-value is $0.0053$ which is $\textit{less than}$ the significance level of $\alpha = 0.05$, we $\textit{reject}$ the null hypothesis.

d) There is evidence that there is a significant difference between LDL cholesterol after two weeks of eating cornflakes versus eating oat bran for two weeks. In fact, LDL cholesterol level seems lower on the oat bran diet.