Date | November 2010 | Marks available | 15 | Reference code | 10N.3sp.hl.TZ0.2 |
Level | HL only | Paper | Paper 3 Statistics and probability | Time zone | TZ0 |
Command term | Find | Question number | 2 | Adapted from | N/A |
Question
The length of time, T, in months, that a football manager stays in his job before he is removed can be approximately modelled by a normal distribution with population mean \(\mu \) and population variance \({\sigma ^2}\). An independent sample of five values of T is given below.
6.5, 12.4, 18.2, 3.7, 5.4
(a) Given that \({\sigma ^2} = 9\),
(i) use the above sample to find the 95 % confidence interval for \(\mu \), giving the bounds of the interval to two decimal places;
(ii) find the smallest number of values of T that would be required in a sample for the total width of the 90 % confidence interval for \(\mu \) to be less than 2 months.
(b) If the value of \({\sigma ^2}\) is unknown, use the above sample to find the 95 % confidence interval for \(\mu \), giving the bounds of the interval to two decimal places.
Markscheme
(a) (i) as \({\sigma ^2}\) is known \({\bar x}\) is \({\text{N}}\left( {\mu ,\frac{{{\sigma ^2}}}{n}} \right)\) (M1)
CI is \(\bar x - {z^ * }\frac{\sigma }{{\sqrt n }} < \mu < \bar x + {z^ * }\frac{\sigma }{{\sqrt n }}\) (M1)
\(\bar x = 9.24,{\text{ }}{z^ * } = 1.960\) for 95 % CI (A1)
CI is \(6.61 < \mu < 11.87\) by GDC A1A1
(ii) CI is \(\bar x - {z^ * }\frac{\sigma }{{\sqrt n }} < \mu < \bar x + {z^ * }\frac{\sigma }{{\sqrt n }}\)
require \(2 \times 1.645\frac{3}{{\sqrt n }} < 2\) R1A1
\(4.935 < \sqrt n \) (A1)
\(24.35 < n\) A1
so smallest value for n = 25 A1
Note: Accept use of table.
[10 marks]
(b) as \({\sigma ^2}\) is not known \({\bar x}\) has the t distribution with v = 4 (M1)(A1)
CI is \(\bar x - {t^ * }\frac{{{s_{n - 1}}}}{{\sqrt n }} < \mu < \bar x + {t^ * }\frac{{{s_{n - 1}}}}{{\sqrt n }}\)
\(\bar x = 9.24,{\text{ }}{s_{n - 1}} = 5.984,{\text{ }}{t^ * } = 2.776\) for 95 % CI (A1)
CI is \(1.81 < \mu < 16.67\) by GDC A1A1
[5 marks]
Total [15 marks]
Examiners report
The 2 confidence intervals were generally done well by using a calculator. Some marks were dropped by not giving the answers to 2 decimal places as required. Weak candidates did not realise that (b) was a t interval. Part (a) (ii) was not as well answered and often it was the first step that was the problem.