| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2010 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample t-test |
| Difficulty | Standard +0.3 This is a straightforward two-part hypothesis testing question requiring standard procedures: (a) a one-sample t-test with given summary statistics, and (b) a chi-squared test for variance. Both parts involve routine application of S4 techniques with clear hypotheses and standard significance levels, making it slightly easier than average for A-level Further Maths statistics. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: \mu = 70\), \(H_1: \mu > 70\) | B1 | B1 both hypotheses using \(\mu\) |
| \(t = \frac{71.2 - 70}{3.4/\sqrt{20}} = 1.58\) | M1 A1 | M1 for \(\frac{71.2-70}{3.4/\sqrt{20}}\); A1 awrt 1.73 |
| Critical value \(t_{19}(5\%) = 1.729\) | B1 | |
| Not significant, insufficient evidence to confirm manufacturer's claim | A1ft | A1 correct conclusion ft their \(t\) value and CV |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: \sigma^2 = 16\), \(H_1: \sigma^2 \neq 16\) | B1 | B1 both hypotheses and 16; accept \(\sigma=4\) and \(\sigma\neq 4\) |
| Test statistic \(\frac{(n-1)s^2}{\sigma^2} = \frac{219.64}{16} = 13.7\ldots\) | M1 A1 | M1 for \(\frac{(19)\times 3.4^2}{16}\); allow \(\frac{(19)\times 3.4^2}{4}\); A1 awrt 13.7 |
| Critical values: \(\chi^2_{19}(5\%)\) upper tail \(= 32.852\), lower tail \(= 8.907\); not significant | B1 B1 | B1 for 32.852; B1 for 8.907 |
| Insufficient evidence to suggest that the variance of miles per gallon of the panther is different from that of the Tiger | A1ft | A1 correct contextual comment; NB those using \(\sigma^2=4\) throughout get B0 M1 A0B1 B1 A1 |
# Question 5:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \mu = 70$, $H_1: \mu > 70$ | B1 | B1 both hypotheses using $\mu$ |
| $t = \frac{71.2 - 70}{3.4/\sqrt{20}} = 1.58$ | M1 A1 | M1 for $\frac{71.2-70}{3.4/\sqrt{20}}$; A1 awrt 1.73 |
| Critical value $t_{19}(5\%) = 1.729$ | B1 | |
| Not significant, insufficient evidence to confirm manufacturer's claim | A1ft | A1 correct conclusion ft their $t$ value and CV |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \sigma^2 = 16$, $H_1: \sigma^2 \neq 16$ | B1 | B1 both hypotheses and 16; accept $\sigma=4$ and $\sigma\neq 4$ |
| Test statistic $\frac{(n-1)s^2}{\sigma^2} = \frac{219.64}{16} = 13.7\ldots$ | M1 A1 | M1 for $\frac{(19)\times 3.4^2}{16}$; allow $\frac{(19)\times 3.4^2}{4}$; A1 awrt 13.7 |
| Critical values: $\chi^2_{19}(5\%)$ upper tail $= 32.852$, lower tail $= 8.907$; not significant | B1 B1 | B1 for 32.852; B1 for 8.907 |
| Insufficient evidence to suggest that the variance of miles per gallon of the panther is different from that of the Tiger | A1ft | A1 correct contextual comment; NB those using $\sigma^2=4$ throughout get B0 M1 A0B1 B1 A1 |
---\begin{enumerate}
\item A car manufacturer claims that, on a motorway, the mean number of miles per gallon for the Panther car is more than 70 . To test this claim a car magazine measures the number of miles per gallon, $x$, of each of a random sample of 20 Panther cars and obtained the following statistics.
\end{enumerate}
$$\bar { x } = 71.2 \quad s = 3.4$$
The number of miles per gallon may be assumed to be normally distributed.\\
(a) Stating your hypotheses clearly and using a $5 \%$ level of significance, test the manufacturer's claim.
The standard deviation of the number of miles per gallon for the Tiger car is 4 .\\
(b) Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether or not there is evidence that the variance of the number of miles per gallon for the Panther car is different from that of the Tiger car.\\
\hfill \mbox{\textit{Edexcel S4 2010 Q5 [11]}}