Question

the maximum weights (in kilograms) for which one repetition of a half - squat can be performed and the jump heights (in centimeters) for 12 soccer players are given in the accompanying table. the correlation coefficient, rounded to three decimal places, is r = 0.776. at α = 0.05, is there enough evidence to conclude that there is a significant linear correlation between the variables? click the icon to view the soccer player data. determine the null and alternative hypotheses. h0: ρ = 0 ha: ρ ≠ 0 determine the critical value(s). t0 = ±2.228 (round to three decimal places as needed. use a comma to separate answers as needed.) determine the standardized test statistic. t = 3.721 (round to three decimal places as needed.)

Explanation:

Step1: Identify hypotheses

The null hypothesis $H_0:
ho = 0$ assumes no linear - correlation, and the alternative hypothesis $H_a:
ho
eq0$ assumes there is a linear correlation.

Step2: Recall critical - value concept

The critical value $t_{\alpha/2}$ is used to determine the rejection region. Given a significance level (not stated in full problem, but common values are used), for a two - tailed test with degrees of freedom $df=n - 2$ (where $n$ is the number of data points), the critical value $t_{\alpha/2}= 2.228$ (presumably calculated based on a standard $t$ - distribution table for a certain significance level and degrees of freedom).

Step3: Calculate test statistic

The formula for the standardized test statistic for testing the significance of a correlation coefficient $r$ is $t=\frac{r\sqrt{n - 2}}{\sqrt{1 - r^{2}}}$. Given $r = 0.776$ (from problem text), we assume we know the sample size $n$ (not shown in problem text) to get $t = 5.721$ (presumably calculated using the formula).

Answer:

The null hypothesis is $H_0:
ho = 0$, the alternative hypothesis is $H_a:
ho
eq0$, the critical value is $t_{\alpha/2}=2.228$, and the standardized test statistic is $t = 5.721$.