Answer to Question #252496 in Statistics and Probability for Cleo

Question

1.     A nutrition store in the mall is selling “Memory Booster,” which is a concoction of herbs and minerals that is intended to improve memory performance, but there is no good reason to think it couldn't possibly do the opposite. To test the effectiveness of the herbal mix, a researcher obtains a sample of 26 participants and asks each person to take the suggested dosage each day for 4 weeks. At the end of the 4-week period, each individual takes a standardized memory test. In the general population, the standardized test is known to have a mean of μ = 50.

Answer 1

Let Memory Scores be:

$X=\{44,49,45,52,53,50,50,48,51,50,46,47,55,54,53,40,52,47,55,56,50,$

$47,49,53,50,51\}$

Null hypothesis:

Those who take Memory Booster will have the same memory score as that of the general population.

Alternative hypothesis:

Those who take Memory Booster will have a significantly different memory score from that of the general population.

$H_0:\mu=50$

$H_a:\mu\neq50$

$\overline{x}=\frac{\sum x_i}{n}=49.88$

$s=\sqrt{\frac{(x_i-\mu)^2}{n-1}}=3.73$

$t=\frac{\overline{x}-\mu}{s/\sqrt{n}}=\frac{49.88-50}{3.73/\sqrt{26}}=0.164$

$df=n-1=25$

critical value for $\alpha=0.05$ :

$t_{crit}=2.060$

Since $t<t_{crit}$ , we can accept the null hypothesis.