Moderate -0.3 This is a standard one-sample z-test hypothesis test with all parameters given (known σ = 0.8, α = 0.10). Students must state hypotheses, calculate the sample mean, compute the z-statistic, find the critical value for a two-tailed test, and reach a conclusion. While it requires multiple steps for 7 marks, it follows a routine procedure taught extensively in A-level statistics with no conceptual challenges or novel problem-solving required, making it slightly easier than average.
It is known that a hospital has a mean waiting time of 4 hours for its Accident and Emergency (A\&E) patients.
After some new initiatives were introduced, a random sample of 12 patients from the hospital's A\&E Department had the following waiting times, in hours.
\(4.25\) \quad \(3.90\) \quad \(4.15\) \quad \(3.95\) \quad \(4.20\) \quad \(4.15\)
\(5.00\) \quad \(3.85\) \quad \(4.25\) \quad \(4.05\) \quad \(3.80\) \quad \(3.95\)
Carry out a hypothesis test at the 10\% significance level to investigate whether the mean waiting time at this hospital's A\&E department has changed.
You may assume that the waiting times are normally distributed with standard deviation 0.8 hours.
[7 marks]
Question 14:
14 | States both hypotheses
correctly for two-tailed test
Accept population mean for | 2.5 | B1 | X = waiting times, in hours
𝐻𝐻 0:𝜇𝜇 = 4
𝐻𝐻1:𝜇𝜇 ≠ 4
= = 4.125
49.5
𝑥𝑥
12
4.125 – 4
Test statistic=
0.8
�
√12
=0.541
Critical value 1.65
0.541 < 1.65
Accept
There is𝐻𝐻 i 0 nsufficient evidence to
suggest that the mean waiting time
at this hospital’s A&E department
has changed
𝜇𝜇
Calculates mean of the sample
PI in equation
Accept = [4.12, 4.13] | 1.1a | M1
Formula𝑥𝑥tes the test statistic or
uses the correct distribution of
their sample mean
PI by correct test statistic value
or calculates probability
or
identifies acceptance region
Condone | 1.1a | M1
Obtains th4e −co4r.r1e2c5t value of the
test statistic [0.519, 0.563]
or
obtains the correct probability
[0.286,0.302] or [0.57, 0.604]
or
obtains the correct acceptance
region of [3.62, 4.38] | 1.1b | A1
Compares their value of test
statistic [0.519, 0.563] with their
critical value [1.64, 1.65]
Allow critical value [-4, 4] except
±0.1 or ±0.05
or
compares their probability
[0.286,0.302] with 0.05 or
compares their probability [0.57,
0.604] with 0.10
or
compares their sample mean
[4.12, 4.13] with their
acceptance region [3.62, 4.38]
Do not allow negative region | 1.1b | B1F
Compares values correctly and
infers is not rejected
CSO
Allow r𝐻𝐻e 0 ference to | 2.2b | A1
Concludes correctl y𝐻𝐻 i 1 n context
that there is insufficient
evidence to suggest that the
mean waiting time at this
hospital’s A&E department has
changed | 3.2a | R1
≠ 4
CSO
Total | 7
Q | Marking Instructions | AO | Marks | Typical Solution
It is known that a hospital has a mean waiting time of 4 hours for its Accident and Emergency (A\&E) patients.
After some new initiatives were introduced, a random sample of 12 patients from the hospital's A\&E Department had the following waiting times, in hours.
$4.25$ \quad $3.90$ \quad $4.15$ \quad $3.95$ \quad $4.20$ \quad $4.15$
$5.00$ \quad $3.85$ \quad $4.25$ \quad $4.05$ \quad $3.80$ \quad $3.95$
Carry out a hypothesis test at the 10\% significance level to investigate whether the mean waiting time at this hospital's A\&E department has changed.
You may assume that the waiting times are normally distributed with standard deviation 0.8 hours.
[7 marks]
\hfill \mbox{\textit{AQA Paper 3 2020 Q14 [7]}}