S3 June 2014 (R) Q4
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 \(\mathrm{H_1}: \mu \gt 160\)
On reporting to his manager, the researcher can only find 4 of the output values. These are shown below
\[168.2 \qquad 157.4 \qquad 173.3 \qquad 161.1\]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. (6)
| Scheme | Marks |
|---|---|
| Sample mean, \(\bar{x} = \frac{660 + \alpha}{5} = 132 + \frac{\alpha}{5}\) | |
| Test statistic, \(z = \dfrac{132 + \frac{\alpha}{5} - 160}{\frac{6}{\sqrt{5}}}\) | M1A1ft |
| Critical z values is 1.6449 | B1 |
| Therefore the test statistic is significant if \(\dfrac{132 + \frac{\alpha}{5} - 160}{\frac{6}{\sqrt{5}}} \gt 1.6449\) | M1 |
| Therefore \(132 + \dfrac{\alpha}{5} - 160 \gt 1.6449 \times \dfrac{6}{\sqrt{5}}\) \(\alpha \gt 5\left(1.6449 \times \dfrac{6}{\sqrt{5}} + 28\right)\) \(\alpha \gt 162.0686493\ldots\) Accept awrt 162.1 | A1 |
| (6 marks) |
Notes
(The sample mean line is printed at the foot of the Question 3 notes in the mark scheme.)
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