S4 June 2017 Q5
5. Jamland and Goodjam are two suppliers of jars of jam. The weights of the jars of jam produced by each supplier can be assumed to be normally distributed with unknown, but equal, variances. A random sample of 20 jars of jam is taken from those supplied by Jamland.
Based on this sample, the 95% confidence interval for the mean weight of a jar of Jamland jam, in grams, is
\[[\,492,\ \ 507\,]\]A random sample of 10 jars of jam is selected from those supplied by Goodjam. The weight of each jar of Goodjam jam, \(y\) grams, is recorded. The results are summarised as follows
\[\bar{y} = 480 \qquad s_y^{\,2} = 280\]Find a 90% confidence interval for the value by which the mean weight of a jar of jam supplied by Jamland exceeds the mean weight of a jar of jam supplied by Goodjam. (11)
| Scheme | Marks |
|---|---|
| \(\bar{x} = \dfrac{492 + 507}{2}\) | M1 |
| \(= 499.5\) | A1cao |
| \(2.093\dfrac{s}{\sqrt{20}} = 7.5\) | M1,B1 |
| \(s = 16.02533\ldots\ \ (s^2 = 256.81..6)\) | A1 |
| \(s_p^{\,2} = \dfrac{19 \times 16.025..^2 + 9 \times 280}{28} = 264.26\ldots\) | M1A1ft |
| \(t_{28}(0.05) = 1.701\) | B1 |
| 90% CI \(= (499.5 - 480) \pm 1.701 \times \sqrt{264.26} \times \sqrt{\dfrac{1}{20} + \dfrac{1}{10}}\) | M1A1ft |
| \(= (8.8,\ 30.2)\) | A1cao |
| (11) | |
| (11 marks) |
Notes
M1 \(\bar{x} = \dfrac{492 + 507}{2}\)
A1 499.5 cao
M1 \(t\text{-value}\dfrac{s}{\sqrt{20}} = 7.5\)
B1 2.093
A1 awrt 16.0 for \(s\) or 257 for \(s^2\)
M1 \(\dfrac{\text{"}n_1 - 1\text{"} \times \left(s \text{ or } s^2\right) + \text{"}n_2 - 1\text{"} \times \left(s \text{ or } s^2\right)}{n_1 + n_2 - 2}\) finding \(s_p^{\,2}\)
A1 ft their \(s^2\)
B1 awrt 1.701
M1 \((\bar{x} - 480) \pm t\text{-value} \times \sqrt{s_p^{\,2}} \times \sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}\)
A1 ft their \(s_p^{\,2}\) and \(\bar{x}\)
A1cao awrt 8.8 and awrt 30.2