4. A manufacturing company produces solar panels. The output of each solar panel is normally distributed with standard deviation 6 watts. It is thought that the mean output, \(\mu\), is 160 watts. A researcher believes that the mean output of the solar panels is greater than 160 watts. He writes down the output values of 5 randomly selected solar panels. He uses the data to carry out a hypothesis test at the \(5 \%\) level of significance. He tests \(\mathrm { H } _ { 0 } : \mu = 160\) against \(\mathrm { H } _ { 1 } : \mu > 160\) On reporting to his manager, the researcher can only find 4 of the output values. These are shown below $$\begin{array} { l l l l } 168.2 & 157.4 & 173.3 & 161.1 \end{array}$$ Given that the result of the hypothesis test is that there is significant evidence to reject \(\mathrm { H } _ { 0 }\) at the \(5 \%\) level of significance, calculate the minimum possible missing output value, \(\alpha\). Give your answer correct to 1 decimal place.

**4**

| Answer | Marks | Guidance |
|--------|-------|----------|
| Test statistic, $z = \frac{132 + \frac{\alpha}{5} - 160}{\frac{6}{\sqrt{5}}}$ | M1A1ft | |
| Critical $z$ values is 1.6449 | B1 | |
| Therefore the test statistic is significant if $\frac{132 + \frac{\alpha}{5} - 160}{\frac{6}{\sqrt{5}}} > 1.6449$ | M1 | |
| Therefore $132 + \frac{\alpha}{5} - 160 > 1.6449 \times \frac{6}{\sqrt{5}}$ | | |
| $\alpha > 5\left(1.6449 \times \frac{6}{\sqrt{5}} + 28\right)$ | | |
| $\alpha > 162.0686493...$ | A1 | |
| Accept awrt 162.1 | | (6) (6 marks) |

**Guidance Notes:**
- 1st A1 ft on their $\bar{x}$
- 1st B1 given for 1.6449 seen (condone sign)
- 3rd M1 inequality using their test statistic, accept incorrect signs for M1

---

4. A manufacturing company produces solar panels. The output of each solar panel is normally distributed with standard deviation 6 watts. It is thought that the mean output, $\mu$, is 160 watts.

A researcher believes that the mean output of the solar panels is greater than 160 watts. He writes down the output values of 5 randomly selected solar panels. He uses the data to carry out a hypothesis test at the $5 \%$ level of significance.

He tests $\mathrm { H } _ { 0 } : \mu = 160$ against $\mathrm { H } _ { 1 } : \mu > 160$\\
On reporting to his manager, the researcher can only find 4 of the output values. These are shown below

$$\begin{array} { l l l l } 
168.2 & 157.4 & 173.3 & 161.1
\end{array}$$

Given that the result of the hypothesis test is that there is significant evidence to reject $\mathrm { H } _ { 0 }$ at the $5 \%$ level of significance, calculate the minimum possible missing output value, $\alpha$. Give your answer correct to 1 decimal place.\\

\hfill \mbox{\textit{Edexcel S3 2014 Q4 [6]}}