Grip and Leg Strength. In the paper, "Sex Differences in Static Strength and Fatigability in Three Different Muscle Groups" (Research Quarterly for Exercise and Sport, Vol. 61(3). pp. 238-242). J. Misner et al. published results of a study on grip and leg strength of males and females. The following data, in newtons, is based on their measurements of right-leg strength.

Preliminary data analyses indicate that you can reasonably presume leg strength is normally distributed for both males and females and that the standard deviations of leg strength are approximately equal. At the $5\%$significance level, do the data provide sufficient evidence to conclude that the mean right-leg strength of males exceeds that of females? (Note: ${\overrightarrow{x}}_1=2127,s_1=513,f_2=1843$, and $s_2=446$.)

Step by step solution

01

Given Information

To calculate there exists a significant difference between the male and female right-leg strengths.

02

Explanation

As the given data is normally distributed and the given population standard deviations are more or less equal, the pooled t-test can be used as follows:

$H_0:{\mu }_1={\mu }_2$

$H_{\alpha }:{\mu }_1>{\mu }_2$

Here the pooled standard deviation needs to be calculated as follows:

$$\begin{gathered} s\sqrt{\frac{\left(n_1-1\right)s_1^2+{\left(n_2-1\right)}_{s_2}^2}{2}} \\ =\sqrt{\frac{(13-1){513}^2+(14-1){446}^2}{13+14-2}} \\ \approx 479.33 \end{gathered}$$

Now, the test statistic needs to be calculated:

$$\begin{gathered} t=\frac{{\overline{x}}_1-{\overline{x}}_2}{s_1\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \\ =\frac{2127-1843}{479.33\sqrt{\frac{1}{13}+\frac{1}{14}}} \\ \approx 1.538 \end{gathered}$$

Now, the P-value corresponding to the test statistic needs to be determined with

$$\begin{gathered} df=n_1+n_2-2 \\ =13+14-2 \\ =25 \\ 0.05<P<0.10 \end{gathered}$$

Now as the P-value is greater than the significance level, the null hypothesis is not rejected

$$\begin{gathered} \mathrm{P}>0.05\Rightarrow \\ \text{Do not reject}{\mathrm{H}}_0 \end{gathered}$$

Therefore, there is not a significant difference between the two datasets.

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