AQA Further Paper 3 Statistics 2024 June — Question 6 9 marks

Exam BoardAQA
ModuleFurther Paper 3 Statistics (Further Paper 3 Statistics)
Year2024
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicT-tests (unknown variance)
TypeSingle sample t-test
DifficultyStandard +0.3 This is a straightforward application of a one-sample t-test with all necessary summary statistics provided. Students must state the normality assumption and perform a standard hypothesis test procedure. The calculations are routine (finding sample mean, standard deviation, test statistic, and comparing to critical value), requiring no novel insight beyond textbook methodology. Slightly easier than average due to small sample size simplifying arithmetic and clear structure.
Spec5.05c Hypothesis test: normal distribution for population mean
6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6
  1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
  2. Investigate Imran's claim, using the 10\% level of significance.
Question 6(a):
AnswerMarks Guidance
Assume that retirement ages are normally distributedB1 States normally distributed OE
Question 6(b):
Hypotheses
AnswerMarks Guidance
\(H_0: \mu = 29.5\), \(H_1: \mu \neq 29.5\)B1 States both hypotheses using correct language
Sample statistics
AnswerMarks
\(\bar{x} = 30.42\), \(s^2 = 1.9525\) OE; \(s^2\) implied by \(s = \dfrac{\sqrt{781}}{20}\) or \(\mathbf{AWRT}\ 1.40\)B1
Test statistic
AnswerMarks Guidance
\(t = \dfrac{\bar{x} - 29.5}{\sqrt{\dfrac{s^2}{5}}}\); condone \(z =\)M1 Attempts to calculate \(t = \dfrac{\text{their}\ \bar{x} - 29.5}{\sqrt{\dfrac{\text{their}\ s^2}{5}}}\)
\(t = \dfrac{30.42 - 29.5}{\sqrt{\dfrac{1.9525}{5}}} = \mathbf{AWRT}\ 1.47\)A1
Critical value
AnswerMarks
\(t_4\) at \(95\% = \mathbf{AWRT}\ 2.13\) or \(p = \mathbf{AWRT}\ 0.11\)B1
Comparison
AnswerMarks Guidance
\(1.47 < 2.13\)M1 Evaluates \(t\) model by correctly comparing test statistic with critical value, or \(p\) value with 0.05 (or 0.1 one-tailed), or via confidence interval
Inference
AnswerMarks Guidance
Do not reject \(H_0\)A1F FT their comparison using \(t\) model; condone Accept \(H_0\) or Reject \(H_1\)
Contextual conclusion
AnswerMarks Guidance
Insufficient evidence to suggest that the mean retirement age is not 29.5R1 Must not be definite; must refer to mean retirement age 
## Question 6(a):

Assume that retirement ages are normally distributed | B1 | States normally distributed OE

---

## Question 6(b):

**Hypotheses**
$H_0: \mu = 29.5$, $H_1: \mu \neq 29.5$ | B1 | States both hypotheses using correct language

**Sample statistics**
$\bar{x} = 30.42$, $s^2 = 1.9525$ OE; $s^2$ implied by $s = \dfrac{\sqrt{781}}{20}$ or $\mathbf{AWRT}\ 1.40$ | B1 |

**Test statistic**
$t = \dfrac{\bar{x} - 29.5}{\sqrt{\dfrac{s^2}{5}}}$; condone $z =$ | M1 | Attempts to calculate $t = \dfrac{\text{their}\ \bar{x} - 29.5}{\sqrt{\dfrac{\text{their}\ s^2}{5}}}$

$t = \dfrac{30.42 - 29.5}{\sqrt{\dfrac{1.9525}{5}}} = \mathbf{AWRT}\ 1.47$ | A1 |

**Critical value**
$t_4$ at $95\% = \mathbf{AWRT}\ 2.13$ or $p = \mathbf{AWRT}\ 0.11$ | B1 |

**Comparison**
$1.47 < 2.13$ | M1 | Evaluates $t$ model by correctly comparing test statistic with critical value, or $p$ value with 0.05 (or 0.1 one-tailed), or via confidence interval

**Inference**
Do not reject $H_0$ | A1F | FT their comparison using $t$ model; condone Accept $H_0$ or Reject $H_1$

**Contextual conclusion**
Insufficient evidence to suggest that the mean retirement age is not 29.5 | R1 | Must not be definite; must refer to mean retirement age
6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old.

Imran claims that the mean retirement age is no longer 29.5 years old.\\
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, $x$. The results are

$$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$

6
\begin{enumerate}[label=(\alph*)]
\item State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim.

6
\item Investigate Imran's claim, using the 10\% level of significance.
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2024 Q6 [9]}}
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