Question

The Westbrook Pharmaceutical Company manufactures atorvastatin pills, designed to lower cholesterol levels. Listed below are the amounts (in mg) of atorvastatin in a random sample of the pills. Use a 0.05 significance level to test the claim that the pills come from a population in which the amount of atorvastatin is equal to 25mg.
24.1, 24.4, 24.3, 24.9, 24.1, 26.2, 25.1, 24.7, 24.4, 25.0, 24.7, 25.1, 25.3, 25.5, 25.5

Answer 1

Solution :

Given :

$\mu = 24$

n = 15

$\overline x = 24.89$

s = 0.5902

The hypothesis :

$H_0 : \mu = 25$

$H_a : \mu \neq 25$

This is a 2 tailed test.

The significance level is $95 \% \ ( \alpha =0.05)$

The test statistic :

$Z=\frac{\overline x - \mu}{\frac{\sigma}{\sqrt n}}$

$Z=\frac{24.89-25}{\frac{0.5902}{\sqrt 15}}$

   = -0.72

The p value : The p value for Z = -0.72 is 0.4716

The critical value : the critical value at α = 0.05 is +1.96 to -1.96

The decision rule :

If $Z_{observed} > Z_{critical}$ or if $Z_{observed} < -Z_{critical}$, then reject $H_0$.

Also if p value is less than α, then reject $H_0$.

The decision :

Since the Z falls in between +1.96 and -1.96, we fail to reject the $H_0$. Also since p value is greater than α, we fail to reject  $H_0$.

The conclusion :

There is not sufficient evidence at the 95% significance level to warrant rejection of the claim that the pills come from a population in which the amount of the atorvastatin is equal to 25 mg.

Now calculating the mean and the standard deviation :

$\text{Mean} = \frac{\text{sum of observation}}{\text{total observations}}$

Standard deviation = $\sqrt{\text{variance}}$

Variance = $\frac{\text{sum of squares(SS)}}{n-1}$

Where, SS = $\sum (X - \text{mean})^2$

X                   Mean                 $(X-\text{mean})^2$

24.1               24.89                    0.62

24.4              24.89                     0.24

24.3              24.89                    0.35

24.9             24.89                       0

24.1              24.89                    0.62

24.2             24.89                    1.72

24.1              24.89                    0.04

26.2             24.89                    0.04

25.1              24.89                   0.24

25                24.89                    0.01

24.7             24.89                    0.04

25.1             24.89                    0.04

25.3             24.89                   0.17

25.5             24.89                   0.37

25.5            24.89                   0.37

n                                                   15

Sum                                           373.3

Average                                    24.89

SS                                              4.8775

Variance = $\frac{SS}{n-1}$                       0.348392857

Standard deviation                    0.5902