Let X1, X2, X3, X4 equal the cholesterol level of a woman under the age of 50, a man

QUESTION:

Let \(X_{1}, X_{2}, X_{3}, X_{4}\) equal the cholesterol level of a woman under the age of 50, a man under 50, a woman 50 or older, and a man 50 or older, respectively. Assume that the distribution of \(X_{i}\) is \(N\left(\mu_{i}, \sigma^{2}\right), i=1,2,3,4\). We shall test the null hypothesis \(H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}\), using seven observations of each \(X_{i}\).

(a) Give a critical region for an \(\alpha=0.05\) significance level.

(b) Construct an ANOVA table and state your conclusion, using the following data:

\(\begin{array}{llllllll}
x_{1}: & 221 & 213 & 202 & 183 & 185 & 197 & 162 \\
x_{2}: & 271 & 192 & 189 & 209 & 227 & 236 & 142 \\
x_{3}: & 262 & 193 & 224 & 201 & 161 & 178 & 265 \\
x_{4}: & 192 & 253 & 248 & 278 & 232 & 267 & 289
\end{array}\)

(c) Give bounds on the \(p \text {-value }\) for this test.

(d) For each set of data, construct box-and-whisker diagrams on the same figure and give an interpretation of your diagram.