Chapter PV, Problem 22RE

(a)

To determine

To find out what are researchers’ null and alternative hypotheses.

Answer to Problem 22RE

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1\ne p_2\end{array}$

Explanation of Solution

It is given in the question that a recent study of perfect pitch tested students in American music conservatories. Also, it is given,

  $\begin{array}{l}{\widehat{p}}_1=7\%=0.07\\ {\widehat{p}}_2=32\%=0.32\\ P<0.0001\end{array}$

  $\alpha =0.05$

Thus, the given claim in the question is the difference in proportions. The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population proportions are equal. If the null hypothesis is the claim, then the alternative hypothesis states the opposite of the null hypothesis. So, we have,

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1\ne p_2\end{array}$

(b)

To determine

To state your conclusion.

Explanation of Solution

It is given in the question that a recent study of perfect pitch tested students in American music conservatories. Also, it is given,

  $\begin{array}{l}{\widehat{p}}_1=7\%=0.07\\ {\widehat{p}}_2=32\%=0.32\\ P<0.0001\end{array}$

  $\alpha =0.05$

Now, as we know that if the P-value is smaller than the significance level, then reject the null hypothesis. So, we have,

  $P<0.05\Rightarrow \text{ Reject }{H}_0$

Thus, there is sufficient evidence to support the claim that there is a difference in proportions of perfect pitches between Asian and non-Asian students.

(c)

To determine

To explain in this context what the P-value means.

Explanation of Solution

It is given in the question that a recent study of perfect pitch tested students in American music conservatories. Also, it is given,

  $\begin{array}{l}{\widehat{p}}_1=7\%=0.07\\ {\widehat{p}}_2=32\%=0.32\\ P<0.0001\end{array}$

  $\alpha =0.05$

  $\begin{array}{l}{\widehat{p}}_2-{\widehat{p}}_1=0.32-0.07\\ =0.25\\ =25\%\end{array}$

Thus, we know that the P-value is the probability of obtaining the value of the test statistics or a value more extreme, assuming that the null hypothesis is true. When there is no difference in the proportions of perfect pitches, then we have a less than $0.01\%$ chance to obtain a difference in sample proportions of $25\%$ or more.

(d)

To determine

To find out do you agree or not and why or why not.

Answer to Problem 22RE

No, we do not agree.

Explanation of Solution

It is given in the question that a recent study of perfect pitch tested students in American music conservatories. Also, it is given,

  $\begin{array}{l}{\widehat{p}}_1=7\%=0.07\\ {\widehat{p}}_2=32\%=0.32\\ P<0.0001\end{array}$

  $\alpha =0.05$

Thus, for the claim given in the question, we do not agree with that claim because the study is not an experiment as no treatments were assigned and then it is likely that the cause is another variable such as length, weight, etc.