Standard +0.3 This is a standard one-tailed hypothesis test using the binomial distribution with clearly stated parameters (n=25, p=0.7, significance level 2.5%). Students must find P(X≥21) and compare to 0.025, requiring competent use of binomial probabilities but following a routine procedure with no conceptual challenges beyond A-level statistics curriculum expectations.
A council found that 70% of its new local businesses made a profit in their first year.
The council introduced an incentive scheme for its residents to encourage the use of new local businesses.
At the end of the scheme, a random sample of 25 new local businesses was selected and it was found that 21 of them had made a profit in their first year.
Using a binomial distribution, investigate, at the 2.5% level of significance, whether there is evidence of an increase in the proportion of new local businesses making a profit in their first year.
[6 marks]
Question 17:
17 | States both hypotheses
correctly for one-tailed test
0.7 OE | 2.5 | B1 | : p = 0.7
: p > 0.7
H0
UH1
nder null hypothesis
X B(25, 0.7)
P( ∼X ≥ 21) = 1 – P(X ≤ 20)
= 1 – 0.9095
= 0.0905
0.0905 > 0.025
Do not reject H
0
There is insufficient evidence of an
increase in the proportion of local
businesses that made a profit in
their first year.
States or uses correct model
PI by calculation of one of the
probabilities below
P(X ≤ 19) = [0.806, 0.807]
P(X ≤ 20) = [0.909, 0.91]
P(X ≤ 21) = [0.966, 0.967]
P(X ≥ 20) = [0.193, 0.1935]
P(X ≥ 21) = [0.09, 0.091]
P(X ≥ 22) = [0.033, 0.0333]
P(X ≥ 23) = [0.0089, 0.00896]
or critical value of 23 or
critical region ≥23
condone missing or incorrect
labels | 3.3 | M1
Obtains [0.09, 0.091] or
[0.909, 0.91]
or
obtains critical value 23 or
critical region ≥ 23 | 1.1b | A1
Evaluates binomial model by
correctly comparing their
P(X ≥ 21) or [0.09, 0.091] with
0.025
or
evaluates binomial model by
correctly comparing their
P(X < 21) with 0.975
or evaluates binomial model by
correctly determining if 21 is in
their critical region | 3.5a | M1
Infers H or null hypothesis not
0
rejected
Condone H accepted
0
All figures must be correct
Ignore reference to H
1 | 2.2b | A1
Concludes correctly in context
that there is insufficient
evidence of an increase in the
proportion of local businesses
that made a profit in their first
year.
To be awarded R1, marks
M1A1M1A1 must be scored as
the minimum
Labels of probability calculations
must be correct
Conclusion must not be definite | 3.2a | R1
Question 17 Total | 6
Question Paper Total | 100
A council found that 70% of its new local businesses made a profit in their first year.
The council introduced an incentive scheme for its residents to encourage the use of new local businesses.
At the end of the scheme, a random sample of 25 new local businesses was selected and it was found that 21 of them had made a profit in their first year.
Using a binomial distribution, investigate, at the 2.5% level of significance, whether there is evidence of an increase in the proportion of new local businesses making a profit in their first year.
[6 marks]
\hfill \mbox{\textit{AQA Paper 3 2023 Q17 [6]}}