| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Two-tail z-test |
| Difficulty | Moderate -0.3 This is a straightforward two-tail z-test with given summary statistics. Students must calculate sample mean and standard deviation, then apply the standard hypothesis test procedure at 1% significance level. Part (ii) tests understanding of CLT, which is routine bookwork. The calculations are mechanical with no conceptual challenges beyond standard S2 content. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\bar{x} = 43.5/100 = 0.435\) | B1 | |
| \(s = \sqrt{\dfrac{100}{99} \times \left(\dfrac{31.56}{100} - 0.435^2\right)} = 0.3573\) | M1 | \(s = \sqrt{\dfrac{31.56}{100} - 0.435^2}\) M0; \((= 0.3555)\), or Var \((= 0.126)\) |
| or Var \((= 0.128)\) or \(\frac{1}{99}(31.56 - (43.5)^2/100)\) | ||
| \(H_0\): Pop mean (for B) \(= 0.336\) | B1 | Undefined mean: B0, but allow just "\(\mu\)" |
| \(H_1\): Pop mean (for B) \(\neq 0.336\) | ||
| \(\dfrac{0.435 - 0.336}{\frac{0.3573}{\sqrt{100}}}\) | M1 | Or \(x_\text{crit} = 0.336 \pm 2.576\sqrt{0.12765/100}\) |
| \(= 2.77\) (3 sfs) | A1 | Or \(x_\text{crit} = (0.244)\) or \(0.428\); \(z = 2.785\) (3 sfs) A0 |
| \(Z_\text{crit} = 2.576\) | B1 | Or use of area — correct 0.005 (2-tail) or 0.01 (1-tail) |
| (or 2.326 consistent with 1-tail test) | ||
| Valid comparison with \(z\)-value | M1 | Valid comp \(P(z > 2.77)\) with 0.005 or 0.01; or comp 0.435 with "0.428" |
| Evidence that B amounts differ from A | A1ft [8] | No errors seen. Conclusion consistent with their \(H_0/H_1\). No contradictions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Must state or imply "No" to score these marks | ||
| \(n\) large | B1 | |
| \(\bar{X}\) approx normally distributed or CLT applies | B1 [2] | B0 for "No" with invalid (or no) reason. SR both reasons correct but wrong conclusion scores SR B1 |
## Question 6:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\bar{x} = 43.5/100 = 0.435$ | B1 | |
| $s = \sqrt{\dfrac{100}{99} \times \left(\dfrac{31.56}{100} - 0.435^2\right)} = 0.3573$ | M1 | $s = \sqrt{\dfrac{31.56}{100} - 0.435^2}$ M0; $(= 0.3555)$, or Var $(= 0.126)$ |
| or Var $(= 0.128)$ or $\frac{1}{99}(31.56 - (43.5)^2/100)$ | | |
| $H_0$: Pop mean (for B) $= 0.336$ | B1 | Undefined mean: B0, but allow just "$\mu$" |
| $H_1$: Pop mean (for B) $\neq 0.336$ | | |
| $\dfrac{0.435 - 0.336}{\frac{0.3573}{\sqrt{100}}}$ | M1 | Or $x_\text{crit} = 0.336 \pm 2.576\sqrt{0.12765/100}$ |
| $= 2.77$ (3 sfs) | A1 | Or $x_\text{crit} = (0.244)$ or $0.428$; $z = 2.785$ (3 sfs) A0 |
| $Z_\text{crit} = 2.576$ | B1 | Or use of area — correct 0.005 (2-tail) or 0.01 (1-tail) |
| (or 2.326 consistent with 1-tail test) | | |
| Valid comparison with $z$-value | M1 | Valid comp $P(z > 2.77)$ with 0.005 or 0.01; or comp 0.435 with "0.428" |
| Evidence that B amounts differ from A | A1ft [8] | No errors seen. Conclusion consistent with their $H_0/H_1$. No contradictions |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Must state or imply "No" to score these marks | | |
| $n$ large | B1 | |
| $\bar{X}$ approx normally distributed or CLT applies | B1 [2] | B0 for "No" with invalid (or no) reason. SR both reasons correct but wrong conclusion scores SR B1 |6 A clinic monitors the amount, $X$ milligrams per litre, of a certain chemical in the blood stream of patients. For patients who are taking drug $A$, it has been found that the mean value of $X$ is 0.336 . A random sample of 100 patients taking a new drug, $B$, was selected and the values of $X$ were found. The results are summarised below.
$$n = 100 , \quad \Sigma x = 43.5 , \quad \Sigma x ^ { 2 } = 31.56 .$$
(i) Test at the $1 \%$ significance level whether the mean amount of the chemical in the blood stream of patients taking drug $B$ is different from that of patients taking drug $A$.\\
(ii) For the test to be valid, is it necessary to assume a normal distribution for the amount of chemical in the blood stream of patients taking drug $B$ ? Justify your answer.
\hfill \mbox{\textit{CAIE S2 2010 Q6 [10]}}