A claim is made that there is no correlation between taste and price in french vanilla cappuccino. A sample of 9 brands of french vanilla cappuccino were ranked according to their taste and their price. These ranks are given in the table below.
\begin{tabular}{|r|r|r|}
\hline Brand & Taste & Price \\
\hline A & 2 & 1 \\
\hline B & 9 & 9 \\
\hline C & 4 & 7 \\
\hline D & 6 & 3 \\
\hline E & 8 & 6 \\
\hline F & 3 & 2 \\
\hline G & 7 & 5 \\
\hline H & 1 & 8 \\
\hline I & 5 & 4 \\
\hline
\end{tabular}
Round your answers to 3 places after the decimal point, if necessary.
(a) Find the value of the (Spearman's) rank correlation coefficient test statistic.
Test statistic: $r_{s}=$
(b) Find the critical values of (Spearman's) rank correlation coefficient if this test is conducted at the 0.1 significance level. Do not type " \pm " in front of your answer (the \pm is already given in front of the answer box below).
Critical values: $r_{s}= \pm$
(c) What is the correct conclusion of this test?
There is not sufficient evidence to warrant rejection of the claim that there is no correlation between taste and price.
There is sufficient evidence to support the claim that there is no correlation between taste and price.
There is not sufficient evidence to support the claim that there is no correlation between taste and price.
There is sufficient evidence to warrant rejection of the claim that there is no correlation between taste and price.
Answer
Thus, the value of the (Spearman's) rank correlation coefficient test statistic is \(\boxed{0.35}\).
Step 1 ：Given the taste ranks as [2, 9, 4, 6, 8, 3, 7, 1, 5] and price ranks as [1, 9, 7, 3, 6, 2, 5, 8, 4], we can calculate the differences between the ranks for each brand.
Step 2 ：The differences are calculated as [1, 0, -3, 3, 2, 1, 2, -7, 1].
Step 3 ：We then use the Spearman's rank correlation coefficient formula, which is \(r_s = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}\), where \(d_i\) is the difference in ranks for each item and \(n\) is the number of items.
Step 4 ：Substituting the values into the formula, we get \(r_s = 1 - \frac{6 \sum (1, 0, -3, 3, 2, 1, 2, -7, 1)^2}{9(9^2 - 1)}\).
Step 5 ：Solving the equation, we find that the Spearman's rank correlation coefficient is approximately 0.35.
Step 6 ：Thus, the value of the (Spearman's) rank correlation coefficient test statistic is \(\boxed{0.35}\).
