S3 June 2015 Q4
4. The weights of bags of rice, \(X\) kg, have a normal distribution with unknown mean \(\mu\) kg and known standard deviation \(\sigma\) kg. A random sample of 100 bags of rice gave a 90% confidence interval for \(\mu\) of (0.4633, 0.5127).
State your hypotheses clearly and write down the significance level you have used. (3)
A second random sample, of 150 of these bags of rice, had a mean weight of 0.479 kg.
| Scheme | Marks |
|---|---|
| \(\mathrm{H_0}\): \(\mu = 0.5 \quad\)\(\mathrm{H_1}\): \(\mu \neq 0.5\) | B1 |
| (Significance level = )10% | dB1 |
| (0.5 is in the interval so not significant, accept \(\mathrm{H_0}\), can accept) \(\mu = 0.5\) | B1 |
| (3) |
Notes
1st B1 for both hypotheses in terms of \(\mu\).
2nd dB1 for 10% but accept 5% if they have a one-tail test as \(\mathrm{H_1}\)
3rd B1 for a correct comment leading to accepting \(\mathrm{H_0}\)
Ignore any ‘further calculations’.
| Scheme | Marks |
|---|---|
| \(1.6449 \times \dfrac{\sigma}{\sqrt{100}} = 0.0247\) | M1 B1 |
| \(\sigma = 0.15016\) or \(\dfrac{10 \times 0.0247}{1.6449}\) (awrt 0.15) | A1 |
| \(0.479 \pm 1.96 \times \dfrac{\text{"}\sigma\text{"}}{\sqrt{150}}\) | M1 B1 |
| awrt (0.455, 0.503) | A1 |
| (6) | |
| (9 marks) |
Notes
1st M1 for \(z\dfrac{\sigma}{\sqrt{100}} = k\), using \(n = 100\) and where \(|z| \gt 1.5\) and \(0.02 \lt k \lt 0.03\)
1st B1 for 1.6449 or better in an attempt (could be \(1.6449\sigma = k\) or even \(1.6449\ \sigma^2 = k\))
1st A1 for a correct expression for \(\sigma\) e.g. awrt 0.15
2nd M1 for \(\bar{x} \pm z \times \dfrac{\sigma}{\sqrt{150}}\) for any \(z\) (> 1) and ft their \(\sigma\) and allow \(\bar{x} \in (0.4633, 0.5127)\)
Allow use of letter \(\sigma\) without a value.
2nd B1 for 1.96 or better in an attempt (could be \(1.96\sigma\) or even \(1.96\ \sigma^2\))
2nd A1 for awrt 0.455 and awrt 0.503