Chapter 7, Problem 1P

The mean serum-creatinine level measured in 12 patients 24 hours after they received a newly proposed antibiotic was 1.2 mg/dL.

If the mean and standard deviation of serum creatinine in the general population are 1.0 and 0.4 mg/dL, respectively, then, using a significance level of .05, test whether the mean serum-creatinine level in this group is different from that of the general population.

To determine

Check whether the mean serum-creatine level of the patients is different from that of the general population.

Answer to Problem 1P

There is no sufficient evidence that the mean serum-creatine level of the patients is different from that of the general population at 5% level of significance.

Explanation of Solution

The mean serum-creatine level of the patients is 1.2 mg/dL and the population standard deviation is 0.4 mg/dL. A sample of 12 patients is selected.

The null hypothesis is as follows:

$H_0:\mu =1.0$

The mean serum-creatine level of the patients is not different from that of the general population.

The alternative hypothesis is as follows:

$H_1:\mu \ne 1.0$

The mean serum-creatine level of the patients is different from that of the general population.

Test statistic:

The test statistic is calculated below:

$\begin{array}{c}z=\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}\\ =\frac{1.2-1}{\frac{0.4}{\sqrt{12}}}\\ =1.732\end{array}$

Critical value:

From Standard normal distribution table, the z critical values corresponding to $0.975\left(=1-\frac{0.05}{2}\right)$is $\pm 1.96$.

Decision rule based on critical value approach:

If $z\le –z_{\left(\frac{\alpha }{2}\right)}\left(or\right)z\ge z_{\left(\frac{\alpha }{2}\right)}$, then reject the null hypothesis H₀.

If $–z_{\left(\frac{\alpha }{2}\right)}\le z\le z_{\left(\frac{\alpha }{2}\right)}$, then fail to reject the null hypothesis H_0.

Conclusion based on critical value approach:

The test statistic value is given as 1.732 and critical value is $\pm 1.96$.

That is, $–1.96<1.732<1.96$, the test statistic lies between –1.96 and 1.96.

By the rejection rule, do not reject the null hypothesis H₀.

There is no sufficient evidence that the mean serum-creatine level of the patients is different from that of the general population at 5% level of significance.