| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Calculate Type I error probability |
| Difficulty | Standard +0.8 This S4 hypothesis testing question requires understanding of Type I/II errors, power functions, and critical evaluation of test procedures. While the binomial calculations are straightforward, students must correctly interpret the two-tailed nature of the test, compute complementary probabilities for power, and provide statistical commentary on test suitability—requiring deeper conceptual understanding beyond routine application. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| P(cure) | 0.2 | 0.3 | 0.4 | 0.5 |
| P(Type II error) | 0.5880 | \(r\) | 0.8565 | \(s\) |
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 drug is claimed to produce a cure to a certain disease in 35\% of people who have the disease. To test this claim a sample of 20 people having this disease is chosen at random and given the drug. If the number of people cured is between 4 and 10 inclusive the claim will be accepted. Otherwise the claim will not be accepted.
\begin{enumerate}[label=(\alph*)]
\item Write down suitable hypotheses to carry out this test.
[2]
\item Find the probability of making a Type I error.
[3]
The table below gives the value of the probability of the Type II error, to 4 decimal places, for different values of $p$ where $p$ is the probability of the drug curing a person with the disease.
\begin{tabular}{|c|c|c|c|c|}
\hline
P(cure) & 0.2 & 0.3 & 0.4 & 0.5 \\
\hline
P(Type II error) & 0.5880 & $r$ & 0.8565 & $s$ \\
\hline
\end{tabular}
\item Calculate the value of $r$ and the value of $s$.
[3]
\item Calculate the power of the test for $p = 0.2$ and $p = 0.4$
[2]
\item Comment, giving your reasons, on the suitability of this test procedure.
[2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 Q6 [12]}}