Moderate -0.3 This is a straightforward one-tailed binomial hypothesis test with clearly stated hypotheses (H₀: p=0.3, H₁: p<0.3). Students need to calculate P(X≤2) under H₀ using binomial probabilities and compare to 5% significance level. While it requires understanding of hypothesis testing framework, the calculation is routine and the sample size is small enough for direct computation. Slightly easier than average due to clear setup and standard procedure.
1 Isaac claims that \(30 \%\) of cars in his town are red. His friend Hardip thinks that the proportion is less than \(30 \%\). The boys decided to test Isaac's claim at the \(5 \%\) significance level and found that 2 cars out of a random sample of 18 were red. Carry out the hypothesis test and state your conclusion. [5]
For finding \(P(0,1,2)\) at least two terms of this sum needed
A1
Correct answer accept 0.06(0)
M1
Comparing with 0.05 must be 0.05
A1 ft
Correct conclusion ft their test statistic – no contradictions
OR Using \(N(0.3, 0.0116)\)
\(H_0: p = 0.3\)
B1
Both hypotheses correct
\(H_1: p < 0.3\)
\(z = 0.111 + 1/36 - 0.3 = -1.49159\)
M1
For attempt at z with or without cc
\(-0.0116\)
A1
For correct z
\(-1.49159 \sim 1.645\)
M1
For comparison
Accept Isaac's claim
A1 ft
Correct conclusion ft their test statistic
OR Using \(N(5.4, 3.78)\)
\(H_0: \mu = 5.4\)
B1
Both hypotheses correct
\(H_1: \mu < 5.4\)
\(z = \frac{2.5 - 5.4}{-\sqrt{3.78}} = -1.49159\)
M1
For attempt at z with or without cc
A1
For correct z
\(-1.49159 \sim 1.645\)
M1
For comparison
Accept Isaac's claim
A1 ft
5
$H_0: p = 0.3$ | B1 | Both hypotheses correct
$H_1: p < 0.3$ | |
$P(0,1,2) = 0.7^{13} + 0.3 \times 0.7^{12} \times _{13}C_1 + 0.3^2 \times 0.7^{10} \times _{13}C_2 = 0.001628 + 0.01256 + 0.04576 = 0.0599$ | M1 | For finding $P(0,1,2)$ at least two terms of this sum needed
| A1 | Correct answer accept 0.06(0)
| M1 | Comparing with 0.05 must be 0.05
| A1 ft | Correct conclusion ft their test statistic – no contradictions
**OR** Using $N(0.3, 0.0116)$ | |
$H_0: p = 0.3$ | B1 | Both hypotheses correct
$H_1: p < 0.3$ | |
$z = 0.111 + 1/36 - 0.3 = -1.49159$ | M1 | For attempt at z with or without cc
$-0.0116$ | A1 | For correct z
$-1.49159 \sim 1.645$ | M1 | For comparison
Accept Isaac's claim | A1 ft | Correct conclusion ft their test statistic
**OR** Using $N(5.4, 3.78)$ | |
$H_0: \mu = 5.4$ | B1 | Both hypotheses correct
$H_1: \mu < 5.4$ | |
$z = \frac{2.5 - 5.4}{-\sqrt{3.78}} = -1.49159$ | M1 | For attempt at z with or without cc
| A1 | For correct z
$-1.49159 \sim 1.645$ | M1 | For comparison
Accept Isaac's claim | A1 ft | 5 | Correct conclusion ft their test statistic
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1 Isaac claims that $30 \%$ of cars in his town are red. His friend Hardip thinks that the proportion is less than $30 \%$. The boys decided to test Isaac's claim at the $5 \%$ significance level and found that 2 cars out of a random sample of 18 were red. Carry out the hypothesis test and state your conclusion. [5]
\hfill \mbox{\textit{CAIE S2 2007 Q1 [5]}}