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The null hypothesis states that the population mean is equal to the value mentioned in the claim. If the null hypothesis is the claim, then the alternative hypothesis states the opposite of the null hypothesis.

\(\begin{array}{l}{H_0}:p = 0\\{H_a}:p \ne 0\end{array}\)

The formula for the value of the test statistic is given by,\(z = \frac{{\hat p - {p_0}}}{{\sqrt {\frac{{{p_0}\left( {1 - {p_0}} \right)}}{n}} }}\).

The sample proportion is calculated by dividing the number of successes by the sample size. \(\hat p = \frac{x}{n}\)

(a)

Let the given be:

\(\begin{array}{c}x = 80\\n = 124\\\alpha = 0.05\end{array}\)

Claim that the proportion is\(\frac{2}{3}\).

Sample proportion:

\(\begin{aligned}{c}\hat p &= \frac{x}{n}\\ &= \frac{{80}}{{124}}\\ &\approx 0.6452\end{aligned}\)

The value of the test-statistic:

\(\begin{aligned}{c}z &= \frac{{\hat p - {p_0}}}{{\sqrt {\frac{{{p_0}\left( {1 - {p_0}} \right)}}{n}} }}\\ &= \frac{{0.6452 - 2/3}}{{\sqrt {\frac{{2/3(1 - 2/3)}}{{124}}} }}\\ &\approx - 0.51\end{aligned}\)

When the null hypothesis is true, the P-value is the chance of getting the test statistic's value, or a value that is more extreme. Using the normal probability table in the appendix, calculate the P-value.

\(\begin{aligned}{c}P &= P(Z < - 0.51orZ > 0.51)\\ &= 2P(Z < - 0.51)\\ &= 2(0.3050)\\ &= 0.6100\\ &= 61.00\% \end{aligned}\)

(b)

Since the P-value is smaller than the significance level\(\alpha \), then reject the null hypothesis:

\(P > 0.05 \Rightarrow {\rm{Fail to reject }}{H_0}\)

There isn't enough evidence to back up the allegation that the \(2/3\) of all human beings have a dominant right foot or eye. Hence, the \(2/3\) figure is not implausible.