S2 June 2015 Q2

EdexcelOld spec8 marksIncludes hypothesis testingBinomial Distribution

2. The proportion of houses in Radville which are unable to receive digital radio is 25%.
In a survey of a random sample of 30 houses taken from Radville, the number, \(X\), of houses which are unable to receive digital radio is recorded.

(a) Find \(\mathrm{P}(5 \leqslant X \lt 11)\) (3)

A radio company claims that a new transmitter set up in Radville will reduce the proportion of houses which are unable to receive digital radio. After the new transmitter has been set up, a random sample of 15 houses is taken, of which 1 house is unable to receive digital radio.

(b) Test, at the 10% level of significance, the radio company’s claim. State your hypotheses clearly. (5)

Mark scheme (a)

Scheme	Marks
\(X \sim \mathrm{B}(30, 0.25)\)	B1
\(\mathrm{P}(X \leqslant 10) - \mathrm{P}(X \leqslant 4) = 0.8943 - 0.0979\) \(= 0.7964\)	M1 A1

Notes

B1: using B(30, 0.25)

M1: using \(\mathrm{P}(X \leqslant 10) - \mathrm{P}(X \leqslant 4)\) or \(\mathrm{P}(X \geqslant 5) - \mathrm{P}(X \geqslant 11)\) oe

A1: awrt 0.796

NB a correct answer gains full marks

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : p = 0.25 \quad \mathrm{H}_1 : p \lt 0.25\)	B1
B(15, 0.25) \(\mathrm{P}(X \leqslant 1) = 0.0802\)	M1 A1
Reject \(\mathrm{H}_0\) or Significant or 1 lies in the critical region	dM1
There is evidence that the radio company’s claim is true. Or The new transmitter will reduce the proportion of houses unable to receive radio	A1cso

Notes

B1: Both hypotheses correct, labelled \(\mathrm{H}_0\) or NH or \(\mathrm{H}_n\) and \(\mathrm{H}_1\) or AH or \(\mathrm{H}_a\), must use \(p\) or \(p(x)\) or \(\pi\)

M1: for using B(15, 0.25)

A1: awrt 0.0802 or CR \(X \leqslant 1\) (allow \(\mathrm{P}(X \geqslant 2) = 0.9198\))

NB: Allow M1 A1 for a correct CR with no incorrect working

M1: A correct statement – do not allow contradictory non contextual statements. Follow through their Probability/CR (for 1 or 2 tail test). If no \(\mathrm{H}_1\) given then M0. Ignore their comparison. For a probability < 0.5, statement must be correct compared to 0.1 for 1 tail test and 0.05 for 2 tailed test or if the probability > 0.5, statement must be correct compared to 0.9 for 1 tail test and 0.95 for 2 tailed test.

A1: cso (all previous marks awarded) and a correct statement containing the word company if writing about the claim or radio if full context.