Question

(21.23) A study of the effect of exposure to color (red or blue) on the ability to solve puzzles used 42 subjects. Half the subjects (21) were asked to solve a series of puzzles while in a red-colored environment. The other half (21) were asked to solve the same series of puzzles while in a blue-colored environment. The time taken to solve the puzzles was recorded for each subject. The 21 subjects in the red-colored environment had a mean time for solving the puzzles of 9.64 seconds with standard deviation 3.43; the 21 subjects in the blue-colored environment had a mean time of 15.84 seconds with standard deviation 8.65. The two-sample t statistic for comparing the population means has value

Answer 1

Answer:

There is enough statistical evidence to say that the means are different.

Step-by-step explanation:

The question is incomplete:

"The two-sample t statistic for comparing the population means has value (±0.001)."

The null and alternative hypothesis are:

$H_0: \mu_r-\mu_b=0\\\\H_1: \mu_r-\mu_b\neq0$

The level of significance is assumed to be α=0.01.

The difference of the sample means is:

$M_d=M_r-M_b=9.64-15.84=-6.20$

The standard deviation for the difference of the means is:

$s_d=\sqrt\frac{s_r^2+s_b^2}{n} } =\sqrt\frac{3.43^2+8.65^2}{21} }=\sqrt{\frac{85.59}{21} }=\sqrt{4.12} =2.03$

Then, the t-statistic is:

$t=\frac{M_d-(\mu_r-\mu_b)}{s_d} =\frac{-6.20-0}{2.03}= -3.05$

The degrees of freedom are:

$df=n_r+n_b-2=21+21-2=40$

With 40 degrees of freedom the t-critial for a significance of α=0.01 is t=±2.705.

As the t-statistic lies in the rejection region, the effect is significant and the null hypothesis is rejected.

There is enough statistical evidence to say that the means are different.