S2 January 2013 Q6

EdexcelOld spec10 marksIncludes hypothesis testingBinomial Distribution

(a) Explain what you understand by a hypothesis. (1)

(b) Explain what you understand by a critical region. (2)

Mrs George claims that 45% of voters would vote for her.

In an opinion poll of 20 randomly selected voters it was found that 5 would vote for her.

(c) Test at the 5% level of significance whether or not the opinion poll provides evidence to support Mrs George’s claim. (4)

In a second opinion poll of \(n\) randomly selected people it was found that no one would vote for Mrs George.

(d) Using a 1% level of significance, find the smallest value of \(n\) for which the hypothesis \(\mathrm{H}_0 : p = 0.45\) will be rejected in favour of \(\mathrm{H}_1 : p \lt 0.45\) (3)

Scheme	Marks
A statement concerning a population parameter	B1

Mark scheme (b)

Scheme	Marks
A critical region is the range / set of values / answers or a test statistic or region/area or values (where the test is significant)	B1
that would lead to the rejection of H0 / acceptance of \(\mathrm{H}_1\)	B1
	(3)

Notes

The scheme prints (3) here as the total for parts (a) and (b).

Mark scheme (c)

Scheme	Marks
\(\mathrm{H}_0 : p = 0.45 \qquad \mathrm{H}_1 : p \lt 0.45\) (or \(p \ne 0.45\))
\(X \sim \mathrm{B}(20, 0.45)\)	M1
\(\mathrm{P}(X \leqslant 5) = 0.0553\) CR \(X \leqslant 4\)	A1
Accept \(\mathrm{H}_0\). Not significant. 5 does not lie in the Critical region.	M1d
There is no evidence that the proportion who voted for Mrs George is not 45% or there is evidence to support Mrs George’s claim	A1cso
	(4)

Notes

1st M1 using B(20, 0.45) and finding \(\mathrm{P}(X \leqslant 5)\) or \(\mathrm{P}(X \geqslant 6)\) Using the normal approximation to the binomial is M0

A1 0.0553 (allow 0.9447) if not using CR or CR \(X \leqslant 4\) or \(X \lt 5\)

2nd M1 dependent on previous M being awarded. A correct statement (do not allow if there are contradicting non contextual statements nor award if 2 probabilities are given which would result in different conclusions)

A1cso Conclusion must contain the words Mrs George. There must be no incorrect working seen. If there are no hypotheses you cannot award this mark.

NB A correct contextual statement on it’s own will score M1 A1.

Mark scheme (d)

Scheme	Marks
B(8, 0.45): P(0) = 0.0084	M1
B(7, 0.45): P(0) = 0.0152	A1
Hence smallest value of \(n\) is 8	B1
	(3)
	(10 marks)

Notes

Alternative

Scheme	Marks
\((0.55)^n \lt 0.01\)	M1
\(n\log 0.55 \lt \log 0.01\)
\(n \gt 7.7\ldots\)	A1
Hence smallest value of \(n\) is 8	B1cso

M1 Attempt to find P(0) from B(\(n\), 0.45) or \((0.55)^n \lt 0.01\) or \((0.55)^n = 0.01\) or \((0.55)^n \gt 0.01\)

A1 P(0) = 0.0084 and P(0) = 0.0152 or getting 7.7 May be implied by correct answer.

B1 cso. \(n = 8\) should not come from incorrect working.

NB An answer of 8 on its own with no working gains M1A1B1