Easy -1.2 Part (b) asks only to state hypotheses for a binomial test (H₀: p=0.3, H₁: p≠0.3), which is pure recall of standard hypothesis testing notation with no calculation or problem-solving required. This is simpler than routine computational questions.
5. (a) The discrete random variable \(X \sim \mathrm {~B} ( 40,0.27 )\)
$$\text { Find } \quad \mathrm { P } ( X \geqslant 16 )$$
Past records suggest that \(30 \%\) of customers who buy baked beans from a large supermarket buy them in single tins. A new manager suspects that there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.
(b) Write down the hypotheses that should be used to test the manager's suspicion.
(c) Using a \(10 \%\) level of significance, find the critical region for a two-tailed test to answer the manager's suspicion. You should state the probability of rejection in each tail, which should be less than 0.05
(d) Find the actual significance level of a test based on your critical region from part (c).
One afternoon the manager observes that 12 of the 20 customers who bought baked beans, bought their beans in single tins.
(e) Comment on the manager's suspicion in the light of this observation.
Later it was discovered that the local scout group visited the supermarket that afternoon to buy food for their camping trip.
(f) Comment on the validity of the model used to obtain the answer to part (e), giving a reason for your answer.
\(H_0: p = 0.3 \quad H_1: p \neq 0.3\) (Both correct in terms of \(p\) or \(\pi\))
B1
For both hypotheses in terms of \(p\) or \(\pi\) and \(H_1\) must be 2-tail
Part (c):
Answer
Marks
Guidance
\([Y \sim B(20,\ 0.3)]\) sight of \(P(Y \leqslant 2) = 0.0355\) or \(P(Y \leqslant 9) = 0.9520\)
M1
For correct use of tables to find probability associated with critical value
Critical region is \(\{Y \leqslant 2\}\) (o.e.)
A1
For the correct lower limit of the CR. Do not award for \(P(Y \leqslant 2)\)
\(\{Y \geqslant 10\}\) (o.e.)
A1
For the correct upper limit
Part (d):
Answer
Marks
Guidance
\([0.0355 + (1 - 0.9520)] = 0.0835\) or \(\mathbf{8.35\%}\)
B1ft
ft on their \(0.0355\) and \((1 - \text{their}\ 0.9520)\) provided each probability is less than \(0.05\)
Part (e):
Answer
Marks
Guidance
(Assuming the 20 customers represent a random sample then) 12 is in the CR so the manager's suspicion is supported
B1ft
ft for a comment that relates 12 to their CR and makes a consistent comment relating this to the manager's suspicion
Part (f):
Answer
Marks
Guidance
e.g. (e) requires the 20 customers to be a random sample or independent and the members of the scout group may invalidate this so binomial distribution would not be valid (and conclusion in (e) is probably not valid)
B1
For a comment giving a suitable reason based on lack of independence or the sample not being random so the binomial model is not valid
## Question 5:
### Part (a):
$P(X \geqslant 16) = 1 - P(X \leqslant 15)$ | M1 | For dealing with $P(X \geqslant 16)$ using cumulative probability function
$= 1 - 0.949077\ldots = \text{awrt}\ \mathbf{0.0509}$ | A1 | awrt $0.0509$ (from calculator)
### Part (b):
$H_0: p = 0.3 \quad H_1: p \neq 0.3$ (Both correct in terms of $p$ or $\pi$) | B1 | For both hypotheses in terms of $p$ or $\pi$ and $H_1$ must be 2-tail
### Part (c):
$[Y \sim B(20,\ 0.3)]$ sight of $P(Y \leqslant 2) = 0.0355$ or $P(Y \leqslant 9) = 0.9520$ | M1 | For correct use of tables to find probability associated with critical value
Critical region is $\{Y \leqslant 2\}$ (o.e.) | A1 | For the correct lower limit of the CR. Do not award for $P(Y \leqslant 2)$
$\{Y \geqslant 10\}$ (o.e.) | A1 | For the correct upper limit
### Part (d):
$[0.0355 + (1 - 0.9520)] = 0.0835$ **or** $\mathbf{8.35\%}$ | B1ft | ft on their $0.0355$ and $(1 - \text{their}\ 0.9520)$ provided each probability is less than $0.05$
### Part (e):
(Assuming the 20 customers represent a random sample then) 12 is in the CR so the manager's suspicion is supported | B1ft | ft for a comment that relates 12 to their CR and makes a consistent comment relating this to the manager's suspicion
### Part (f):
e.g. (e) requires the 20 customers to be a random sample or independent and the members of the scout group may invalidate this so binomial distribution would not be valid (and conclusion in (e) is probably not valid) | B1 | For a comment giving a suitable reason based on lack of independence **or** the sample not being random so the binomial model is not valid
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5. (a) The discrete random variable $X \sim \mathrm {~B} ( 40,0.27 )$
$$\text { Find } \quad \mathrm { P } ( X \geqslant 16 )$$
Past records suggest that $30 \%$ of customers who buy baked beans from a large supermarket buy them in single tins. A new manager suspects that there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.\\
(b) Write down the hypotheses that should be used to test the manager's suspicion.\\
(c) Using a $10 \%$ level of significance, find the critical region for a two-tailed test to answer the manager's suspicion. You should state the probability of rejection in each tail, which should be less than 0.05\\
(d) Find the actual significance level of a test based on your critical region from part (c).
One afternoon the manager observes that 12 of the 20 customers who bought baked beans, bought their beans in single tins.\\
(e) Comment on the manager's suspicion in the light of this observation.
Later it was discovered that the local scout group visited the supermarket that afternoon to buy food for their camping trip.\\
(f) Comment on the validity of the model used to obtain the answer to part (e), giving a reason for your answer.
\hfill \mbox{\textit{Edexcel AS Paper 2 Q5 [9]}}