Standard +0.3 This is a straightforward two-sample confidence interval question with known population standard deviations. Students need to recall the formula for the difference of means, calculate the standard error (√(σ²/n₁ + σ²/n₂)), and apply the z-critical value for 90% confidence. All values are given clearly, requiring only direct substitution and arithmetic—slightly easier than average due to its routine nature and clear setup.
The mean Intelligence Quotient (IQ) of a random sample of 15 pupils at School A is 109. The mean IQ of a random sample of 20 pupils at School B is 112. You may assume that the IQs for the populations from which these samples are taken are normally distributed, and that both distributions have standard deviation 15. Find a 90% confidence interval for \(\mu_B - \mu_A\), where \(\mu_A\) and \(\mu_B\) are the population mean IQs. [6]
Use valid formula for C.I.: x B – x A ± zσ √(1/nA + 1/nB) M2
= 112 – 109 ± z 15 √(1/15 + 1/20) A1
= 3 ± 5⋅123 z A1
Use of correct tabular value: z 0.995 = 1⋅64[5] *A1
Answer
Marks
Guidance
C.I. correct to 3 s.f. (dep *A1): 3 ± 8⋅43 or [-5⋅43, 11⋅4] A1
6
[6]
Question 6:
6 | Use valid formula for C.I.: x B – x A ± zσ √(1/nA + 1/nB) M2
= 112 – 109 ± z 15 √(1/15 + 1/20) A1
= 3 ± 5⋅123 z A1
Use of correct tabular value: z 0.995 = 1⋅64[5] *A1
C.I. correct to 3 s.f. (dep *A1): 3 ± 8⋅43 or [-5⋅43, 11⋅4] A1 | 6 | [6]
The mean Intelligence Quotient (IQ) of a random sample of 15 pupils at School A is 109. The mean IQ of a random sample of 20 pupils at School B is 112. You may assume that the IQs for the populations from which these samples are taken are normally distributed, and that both distributions have standard deviation 15. Find a 90% confidence interval for $\mu_B - \mu_A$, where $\mu_A$ and $\mu_B$ are the population mean IQs. [6]
\hfill \mbox{\textit{CAIE FP2 2010 Q6 [6]}}