| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2023 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Two-sample z-test large samples |
| Difficulty | Standard +0.3 This is a standard two-sample t-test with summary statistics provided. Part (a) requires basic calculations of mean and standard deviation from given sums. Part (b) is a routine hypothesis test following a standard procedure taught in S3. Parts (c) and (d) test understanding of test assumptions, which are standard bookwork. The question requires no novel insight—just methodical application of learned techniques with large sample sizes making it straightforward. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Brand | \(\sum x\) | \(\sum \boldsymbol { x } ^ { \mathbf { 2 } }\) | \(\bar { x }\) | \(s\) | Sample size |
| A | 350 | 1753.9744 | 5.0 | 0.24 | 70 |
| B | 331.5 | 1694.65 | \(\alpha\) | β | 65 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\alpha=5.1\) | B1 | cao |
| \(\beta=\sqrt{\frac{1694.65-65\times(5.1)^2}{64}}\) | M1 | For correct method to find \(\beta\) using their \(\alpha\) |
| \(=0.25\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \mu_A=\mu_B\); \(H_1: \mu_A<\mu_B\) | B1 | Both hypotheses correct; must be in terms of \(\mu\) |
| \(z=\pm\dfrac{5.0-5.1}{\sqrt{\dfrac{0.24^2}{70}+\dfrac{0.25^2}{65}}}\) | M1 M1 | First M1 correct SE ft their \(s\) in part (a); second M1 attempt at test statistic ft SE and \(\alpha\) |
| \(=-2.367\ldots\) awrt \(-2.37\) | A1 | Allow 2.37 |
| One-tailed c.v. \(z=-1.6449\) or CR: \(z\leqslant -1.6449\) | B1 | \(-1.6449\) or better (allow 1.6449 if comparing to their 2.37) |
| In CR/Significant/Reject \(H_0\) | M1 | Correct statement ft their CV and test statistic |
| Sufficient evidence to support Roxane's claim (crisps from brand A have lower fat content than brand B) | A1 | Must include words brand A, lower fat, brand B |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Since the sample is large the CLT applies | M1 | Must mention large and CLT |
| No need to assume that fat content is normally distributed | A1 | Context not required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Assumed that \(s^2=\sigma^2\) in both groups | B1 | Sample variance = population variance for both groups |
## Question 6:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\alpha=5.1$ | B1 | cao |
| $\beta=\sqrt{\frac{1694.65-65\times(5.1)^2}{64}}$ | M1 | For correct method to find $\beta$ using their $\alpha$ |
| $=0.25$ | A1 | cao |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \mu_A=\mu_B$; $H_1: \mu_A<\mu_B$ | B1 | Both hypotheses correct; must be in terms of $\mu$ |
| $z=\pm\dfrac{5.0-5.1}{\sqrt{\dfrac{0.24^2}{70}+\dfrac{0.25^2}{65}}}$ | M1 M1 | First M1 correct SE ft their $s$ in part (a); second M1 attempt at test statistic ft SE and $\alpha$ |
| $=-2.367\ldots$ awrt $-2.37$ | A1 | Allow 2.37 |
| One-tailed c.v. $z=-1.6449$ or CR: $z\leqslant -1.6449$ | B1 | $-1.6449$ or better (allow 1.6449 if comparing to their 2.37) |
| In CR/Significant/Reject $H_0$ | M1 | Correct statement ft their CV and test statistic |
| Sufficient evidence to support Roxane's claim (crisps from brand A have lower fat content than brand B) | A1 | Must include words **brand A**, **lower fat**, **brand B** |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Since the sample is **large** the **CLT** applies | M1 | Must mention large and CLT |
| No need to assume that fat content is normally distributed | A1 | Context not required |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Assumed that $s^2=\sigma^2$ in **both** groups | B1 | Sample variance = population variance for **both** groups |\begin{enumerate}
\item Roxane, a scientist, carries out an investigation into the fat content of different brands of crisps.
\end{enumerate}
Roxane took random samples of different brands of crisps and recorded, in grams, the fat content ( $x$ ) of a 30 gram serving.
The table below shows some results for just two of these brands.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
Brand & $\sum x$ & $\sum \boldsymbol { x } ^ { \mathbf { 2 } }$ & $\bar { x }$ & $s$ & Sample size \\
\hline
A & 350 & 1753.9744 & 5.0 & 0.24 & 70 \\
\hline
B & 331.5 & 1694.65 & $\alpha$ & β & 65 \\
\hline
\end{tabular}
\end{center}
(a) Calculate the value of $\alpha$ and the value of $\beta$
Roxane claims that these results show that the crisps from brand A have a lower fat content than the crisps from brand B , as the mean fat content for brand A is, statistically, significantly less than the mean fat content for brand B .\\
(b) Stating your hypotheses clearly, carry out a suitable test, at the $5 \%$ level of significance, to assess Roxane's claim.\\
You should state your test statistic and critical value.\\
(c) For the test in part (b), state whether or not it is necessary to assume that the fat content of crisps is normally distributed. Give a reason for your answer.\\
(d) State an assumption you have made in carrying out the test in part (b).
\hfill \mbox{\textit{Edexcel S3 2023 Q6 [13]}}