| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 11 |
| 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 large sample sizes (n>30), making it slightly easier than average. The question provides all necessary summary statistics, requires straightforward hypothesis setup (one-tailed test), and the calculation follows a routine procedure. The large samples justify the normality assumption easily, and the test statistic calculation is mechanical. This is a textbook application with no novel insight required, making it slightly easier than the typical A-level question. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Group | Sample size | Mean drop in cholesterol (mg/dl) | Standard deviation |
| Special diet | 100 | 75 | 22 |
| Standard diet | 80 | 64 | 31 |
| Answer | Marks |
|---|---|
| \(H_0: \mu_{sp} = \mu_st\); \(H_1: \mu_{sp} > \mu_st\) | B1 B1 |
| \(\alpha = 0.05\); critical region: \(z > 1.6449\) | B1 |
| \(\text{standard error} = \sqrt{\frac{22^2}{100} + \frac{31^2}{80}} = 4.1051\ldots\) | M1 A1 |
| \(z = \frac{75 - 64}{4.1051\ldots} = 2.68\) | M1 A1 |
| Since 2.68 is in the critical region there is evidence to reject \(H_0\) and conclude that the special diet is more effective in reducing blood cholesterol. | M1 A1 \(\checkmark\) |
| Total: 9 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Drop in blood cholesterol levels are normally distributed, or Central Limit Theorem can be applied, or standard deviations of the populations are 22 and 31 | Any two | B1 B1 |
| Total: 2 marks |
## Part (a)
| $H_0: \mu_{sp} = \mu_st$; $H_1: \mu_{sp} > \mu_st$ | B1 B1 |
| $\alpha = 0.05$; critical region: $z > 1.6449$ | B1 |
| $\text{standard error} = \sqrt{\frac{22^2}{100} + \frac{31^2}{80}} = 4.1051\ldots$ | M1 A1 |
| $z = \frac{75 - 64}{4.1051\ldots} = 2.68$ | M1 A1 |
| Since 2.68 is in the critical region there is evidence to reject $H_0$ and conclude that the special diet is more effective in reducing blood cholesterol. | M1 A1 $\checkmark$ |
| **Total: 9 marks** |
## Part (b)
| Drop in blood cholesterol levels are normally distributed, or Central Limit Theorem can be applied, or standard deviations of the populations are 22 and 31 | Any two | B1 B1 |
| **Total: 2 marks** |
---As part of a research project into the role played by cholesterol in the development of heart disease a random sample of 100 patients was put on a special fish-based diet. A different random sample of 80 patients was kept on a standard high-protein low-fat diet. After several weeks their blood cholesterol was measured and the results summarised in the table below.
\begin{tabular}{|c|c|c|c|}
\hline
Group & Sample size & Mean drop in cholesterol (mg/dl) & Standard deviation \\
\hline
Special diet & 100 & 75 & 22 \\
Standard diet & 80 & 64 & 31 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\item Stating your hypotheses clearly and using a 5% level of significance, test whether or not the special diet is more effective in reducing blood cholesterol levels than the standard diet. [9]
\item Explain briefly any assumptions you made in order to carry out this test. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q3 [11]}}