Chapter PVII, Problem 9RE

To determine

To Explain: that this is strong proof of the difference in musical abilities on the basis of handedness yes or not.

Explanation of Solution

Given:

  $\begin{array}{l}{n}_L=53\\ y_L=33\\ n_R=76\\ {\text{y}}_R=38\end{array}$

Formula used:

  ${\widehat{p}}_L=\frac{y_L}{n_L}$

  ${\widehat{p}}_R=\frac{{\text{y}}_R}{n_R}$

  ${\widehat{p}}_{pooled}=\frac{y_L+{\text{y}}_R}{n_L+n_R}$

  $S{E}_{pooled}\left({\widehat{p}}_L-{\widehat{p}}_R\right)=\sqrt{\frac{{\widehat{p}}_{pooled}{\widehat{q}}_{pooled}}{n_L}+\frac{{\widehat{p}}_{pooled}{\widehat{q}}_{pooled}}{n_R}}$

Calculation:

  $H_0:$ There is no difference in the proportion of left-handed and right-handed people with musical abilities.

i.e. $H_0:p_L-p_R=0$

  $H_1:$ There is a difference between the proportion of left-handed and right-handed individuals with musical abilities.

i.e. $H_1:p_L-p_R\ne 0$

Here $p_L$ shows proportion of left handed people and $p_R$ shows proportion of success to right handed people.

  $\begin{array}{c}{\widehat{p}}_L=\frac{y_L}{n_L}\\ =\frac{33}{53}\\ =0.6226\end{array}$

  $\begin{array}{c}{\widehat{p}}_R=\frac{{\text{y}}_R}{n_R}\\ =\frac{38}{76}\\ =0.50\end{array}$

The pooled proportion is,

  $\begin{array}{c}{\widehat{p}}_{pooled}=\frac{y_L+{\text{y}}_R}{n_L+n_R}\\ =\frac{33+38}{53+76}\\ =0.5504\end{array}$

The pooled value to estimate the standard error is,

  $\begin{array}{c}S{E}_{pooled}\left({\widehat{p}}_L-{\widehat{p}}_R\right)=\sqrt{\frac{{\widehat{p}}_{pooled}{\widehat{q}}_{pooled}}{n_L}+\frac{{\widehat{p}}_{pooled}{\widehat{q}}_{pooled}}{n_R}}\\ =\sqrt{\frac{\left(0.5504\right)\left(0.4496\right)}{53}+\frac{\left(0.5504\right)\left(0.4496\right)}{76}}\\ =\sqrt{0.0047+0.0033}\\ =0.089\end{array}$

The observed difference in the proportions is

  $\begin{array}{c}{\widehat{p}}_L-{\widehat{p}}_R\\ =0.6226-0.50\\ =0.1226\end{array}$

The test statistic is,

  $\begin{array}{c}z=\frac{\left({\widehat{p}}_1-{\widehat{p}}_2\right)-0}{S{E}_{pooled}\left({\widehat{p}}_1-{\widehat{p}}_2\right)}\\ =\frac{0.1226}{0.089}\\ =1.38\end{array}$

This is a two-tailed test,therefore the P-value is,

  $\begin{array}{c}P=2P\left(z>1.38\right)\\ =2\left(1-P\left(z\le 1.38\right)\right)\\ =2\left(1-0.9162\right)\\ =0.1676\end{array}$

Since the P-value is very high, null hypotheses are not dismissed. There is not ample evidence to suggest that there is a difference in musical skills between people who are left-handed and right-handed.