In a certain chemical process, it is very important that a particular solution that is to be used as a reactant have a pH of exactly 8.20. A method for determining pH that is available for solutions of this type is known to give measurements that are normally distributed with a mean equal to the actual pH and with a standard deviation of .02. Suppose 10 independent measurements yielded the following pH values:

$$
\begin{matrix} ext{8.18} & ext{8.17}\ ext{8.16} & ext{8.15}\ ext{8.17} & ext{8.21}\ ext{8.22} & ext{8.16}\ ext{8.19} & ext{8.18}\ \end{matrix}
$$

(a) What conclusion can be drawn at the $\alpha=.10$ level of significance? (b) What about at the $\alpha=.5$ level of significance?

Question

In a certain chemical process, it is very important that a particular solution that is to be used as a reactant have a pH of exactly 8.20. A method for determining pH that is available for solutions of this type is known to give measurements that are normally distributed with a mean equal to the actual pH and with a standard deviation of .02. Suppose 10 independent measurements yielded the following pH values:

$$
\begin{matrix} 	ext{8.18} & 	ext{8.17}\ 	ext{8.16} & 	ext{8.15}\ 	ext{8.17} & 	ext{8.21}\ 	ext{8.22} & 	ext{8.16}\ 	ext{8.19} & 	ext{8.18}\ \end{matrix}
$$

(a) What conclusion can be drawn at the $\alpha=.10$ level of significance? (b) What about at the $\alpha=.5$ level of significance?

Quizlet Admin · Accepted Answer

We are given sample of size $n = 10$. Calculate that the sample mean is

$$
\begin{align*}
\hat{\mu} = \frac{1}{n} \sum_{i = 1}^{10}x_i = 8.179
\end{align*}
$$

Define hypothesis

$$
\begin{align*}
H_0: \ \mu = 8.2 \
H_1: \ \mu 
eq 8.2
\end{align*}
$$

where $\mu$ is population mean. We are given that $\sigma = 0.02$ is population standard deviation. We know that test statistics has distribution

$$
\begin{align*}
\sqrt{n} \cdot \frac{\bar{X} - \mu}{\sigma} \Big|_{H_0} \sim AN(0,1)
\end{align*}
$$

The $p$-value of this random sample is

$$
\begin{align*}
p	ext{-value} &= P_{H_0} \left( \Big| \sqrt{n} \cdot \frac{\bar{X} - \mu}{\sigma} \Big| > \Big| \sqrt{n} \cdot \frac{\hat{\mu} - \mu}{\sigma} \Big| ight) 
\
&= P_{H_0} \left( \Big| \sqrt{n} \cdot \frac{\bar{X} - \mu}{\sigma} \Big| > \Big| \sqrt{10} \cdot \frac{8.179- 8.2}{0.02} \Big| ight) 
\
&= P_{H_0} \left( \Big| \sqrt{n} \cdot \frac{\bar{X} - \mu}{\sigma} \Big| > 3.32 ight) = 2(1 - \Phi(3.32))
\
&= 0.001
\end{align*}
$$

where $\Phi$ is CDF of standard normal and we have used table of CDF of standard normal in appendix to obtain final answer. Since $p$-value is less than significance levels in (a) and (b), we reject the null hypothesis in both cases.

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