| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Validity and assumptions questions |
| Difficulty | Standard +0.3 This is a straightforward S4 question on properties of estimators requiring standard techniques: showing unbiasedness using E(aX+bY)=aE(X)+bE(Y), finding variance using Var(aX+bY)=a²Var(X)+b²Var(Y) for independent variables, and minimizing a quadratic by differentiation. All steps are routine applications of formulas with no novel insight required, making it slightly easier than average. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.05b Unbiased estimates: of population mean and variance |
Total
| Answer | Marks | Guidance |
|---|---|---|
| Engineer | A | B |
| January | 17 | 19 |
| July | 19 | 18 |
| Site | Sample size (n) | – |
| Sample mean (x) | Standard deviation (s) | |
| A | 7 | 8.43 |
| B | 13 | 14.31 |
| Answer | Marks | Guidance |
|---|---|---|
| ! | 1.5 | 2 |
| Power | 0.59 | 0.75 |
| Answer | Marks | Guidance |
|---|---|---|
| ! | 1.5 | 2 |
| Power | 0.59 | 0.75 |
| S | c | i |
Question 6:
6
6
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1. George owns a garage and he records the mileage of cars, x thousands of miles, between
services. The results from a random sample of 10 cars are summarised below.
!x = 113.4 !x2 = 1414.08
The mileage of cars between services is normally distributed and George believes that the
standard deviation is 2.4 thousand miles.
Stating your hypotheses clearly, test, at the 5% level of significance, whether or not these
data support George’s belief.
(7)
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Engineer | A | B | C | D | E | F | G | H
January | 17 | 19 | 22 | 26 | 15 | 28 | 18 | 21
July | 19 | 18 | 25 | 24 | 17 | 25 | 16 | 19
Site | Sample size (n) | –
Sample mean (x) | Standard deviation (s)
A | 7 | 8.43 | 4.24
B | 13 | 14.31 | 4.37
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4. A random sample of size 2, X and X , is taken from the random variable X which has a
1 2
continuous uniform distribution over the interval [–a, 2a], a > 0
X + X
(a) Show that X= 1 2 is a biased estimator of a and find the bias.
2
(3)
The random variable Y = kX is an unbiased estimator of a.
(b) Write down the value of the constant k.
(1)
(c) Find Var(Y).
(4)
The random variable M is the maximum of X and X
1 2
The probability density function, m(x), of M is given by
⎧2(x + a)
⎪ −a ! x ! 2a
m(x) = ⎨ 9a2
⎪
⎩ 0 otherwise
(d) Show that M is an unbiased estimator of a.
(4)
3
Given that E(M2) = a2
2
(e) find Var(M).
(1)
(f) State, giving a reason, whether you would use Y or M as an estimator of a.
(2)
A random sample of two values of X are 5 and –1
(g) Use your answer to part (f) to estimate a.
(1)
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! | 1.5 | 2 | 2.5 | 3 | 3.5 | 4
Power | 0.59 | 0.75 | 0.86 | r | 0.96 | 0.97
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! | 1.5 | 2 | 2.5 | 3 | 3.5 | 4
Power | 0.59 | 0.75 | 0.86 | r | 0.96 | 0.97
S | c | i | e | n | ti | s | t’ | s | t | e | st
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(Total 17 marks)
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6. The carbon content, measured in suitable units, of steel is normally distributed. Two
independent random samples of steel were taken from a refining plant at different times
and their carbon content recorded. The results are given below.
Sample A: 1.5 0.9 1.3 1.2
Sample B: 0.4 0.6 0.8 0.3 0.5 0.4
(a) Stating your hypotheses clearly, carry out a suitable test, at the 10% level of
significance, to show that both samples can be assumed to have come from populations
with a common variance &2.
(7)
(b) Showing your working clearly, find the 99% confidence interval for &2 based on both
samples.
(6)
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Turn overA continuous uniform distribution on the interval $[0, k]$ has mean $\frac{k}{2}$ and variance $\frac{k^2}{12}$.
A random sample of three independent variables $X_1$, $X_2$ and $X_3$ is taken from this distribution.
\begin{enumerate}[label=(\alph*)]
\item Show that $\frac{2}{3}X_1 + \frac{1}{2}X_2 + \frac{5}{6}X_3$ is an unbiased estimator for $k$.
[3]
\end{enumerate}
An unbiased estimator for $k$ is given by $\hat{k} = aX_1 + bX_2$ where $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{1}
\item Show that Var($\hat{k}$) = $(a^2 - 2a + 2) \frac{k^2}{6}$
[6]
\item Hence determine the value of $a$ and the value of $b$ for which $\hat{k}$ has minimum variance, and calculate this minimum variance.
[6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 Q6 [15]}}