Standard +0.3 This is a straightforward one-sample t-test with summary statistics provided. Students must calculate sample mean and variance, state the null/alternative hypotheses, perform the t-test, and state the normality assumption. While it requires multiple steps, each is routine for Further Maths Statistics students, making it slightly easier than average overall.
7 A shopkeeper sells chocolate bars which are described by the manufacturer as having an average mass of 45 grams.
The shopkeeper claims that the mass of the chocolate bars, \(X\) grams, is getting smaller on average.
A random sample of 6 chocolate bars is taken and their masses in grams are measured. The results are
$$\sum x = 246 \quad \text { and } \quad \sum x ^ { 2 } = 10198$$
Investigate the shopkeeper's claim using the \(5 \%\) level of significance.
State any assumptions that you make.
Calculates \(t\)-test statistic with their sample mean and variance or confidence interval (with their \(t\) value) (PI); condone absolute value
\(-2.07\) (AWRT); or \(p\) value AWRT \(p = 0.0466\)
A1
Calculates \(t\)-test statistic or confidence interval correctly; condone absolute value
\(t_5\) at \(95\% = 2.015\); \(-2.07 < -2.015\); Reject \(H_0\)
R1
Evaluates \(t\) model by comparing negative test statistic and correct critical value (AWRT \(-2.02\)) or by comparing \(p\) value with \(0.05\) or by comparing sample mean with confidence interval using correct \(t\) value (AWRT 2.02); condone comparing AWRT 2.02 with their positive test statistic
Infers \(H_0\) rejected
E1F
Follow through on their comparison with consistent signs
Some evidence to suggest the shopkeeper is correct that the mass of chocolate bars is getting lighter
E1F
Concludes in context based on their hypotheses (not definite); consistent with decision
# Question 7:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \mu = 45$; $H_1: \mu < 45$ | B1 | States both hypotheses using correct language |
| Assume $X$ can be modelled by a normal distribution | E1 | States required assumption to use a $t$-test |
| $\bar{x} = \dfrac{246}{6} = 41$ | B1 | Calculates sample mean |
| $s^2 = \dfrac{1}{6-1}\left(10198 - \dfrac{246^2}{6}\right) = 22.4$ (AWRT 4.73 for $s$) | B1 | Calculates sample variance or standard deviation |
| $\dfrac{41-45}{\sqrt{\dfrac{22.4}{6}}} = -2.07$ | M1 | Calculates $t$-test statistic with their sample mean and variance or confidence interval (with their $t$ value) (PI); condone absolute value |
| $-2.07$ (AWRT); or $p$ value AWRT $p = 0.0466$ | A1 | Calculates $t$-test statistic or confidence interval correctly; condone absolute value |
| $t_5$ at $95\% = 2.015$; $-2.07 < -2.015$; Reject $H_0$ | R1 | Evaluates $t$ model by comparing negative test statistic and correct critical value (AWRT $-2.02$) or by comparing $p$ value with $0.05$ or by comparing sample mean with confidence interval using correct $t$ value (AWRT 2.02); condone comparing AWRT 2.02 with their positive test statistic |
| Infers $H_0$ rejected | E1F | Follow through on their comparison with consistent signs |
| Some evidence to suggest the shopkeeper is correct that the mass of chocolate bars is getting lighter | E1F | Concludes in context based on their hypotheses (not definite); consistent with decision |
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7 A shopkeeper sells chocolate bars which are described by the manufacturer as having an average mass of 45 grams.
The shopkeeper claims that the mass of the chocolate bars, $X$ grams, is getting smaller on average.
A random sample of 6 chocolate bars is taken and their masses in grams are measured. The results are
$$\sum x = 246 \quad \text { and } \quad \sum x ^ { 2 } = 10198$$
Investigate the shopkeeper's claim using the $5 \%$ level of significance.\\
State any assumptions that you make.\\
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2019 Q7 [9]}}