CAIE S1 2010 November — Question 2 5 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2010
SessionNovember
Marks5
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TopicApproximating Binomial to Normal Distribution
TypeProbability between two values
DifficultyModerate -0.3 This is a straightforward application of the normal approximation to the binomial distribution with continuity correction. Students need to identify n=200, p=2/15, calculate mean and variance, apply continuity correction (21.5 and 34.5), and use standard normal tables. While it requires multiple steps, each is routine and the question clearly signals the method needed.
Spec2.04d Normal approximation to binomial2.04e Normal distribution: as model N(mu, sigma^2)
2 On average, 2 apples out of 15 are classified as being underweight. Find the probability that in a random sample of 200 apples, the number of apples which are underweight is more than 21 and less than 35.
AnswerMarks Guidance
\(\text{mean} = 200 \times 2/15 = 26.67 \text{ (or } 80/3\text{)}\)B1 mean and variance correct
\(\text{variance} = 200 \times 2/15 \times 13/15 = 23.11 \text{ (or } 208/9\text{)}\)
\(P(21 < X < 35) = P\left(\frac{21.5 - 26.67}{\sqrt{23.11}} < z < \frac{34.5 - 26.67}{\sqrt{23.11}}\right)\)M1 standardising, \(\pm\), with or without cc, must have sq rts
\(= P(-1.075 < z < 1.629)\)M1 continuity corrections 20.5 or 21.5, 34.5 or 35.5
\(= 0.8589 + 0.9483 - 1 = 0.807\)M1 \(\Phi_1 + \Phi_2 - 1\)
A1answer rounding to 0.807
[5] 
$\text{mean} = 200 \times 2/15 = 26.67 \text{ (or } 80/3\text{)}$ | B1 | mean and variance correct
$\text{variance} = 200 \times 2/15 \times 13/15 = 23.11 \text{ (or } 208/9\text{)}$ |

$P(21 < X < 35) = P\left(\frac{21.5 - 26.67}{\sqrt{23.11}} < z < \frac{34.5 - 26.67}{\sqrt{23.11}}\right)$ | M1 | standardising, $\pm$, with or without cc, must have sq rts

$= P(-1.075 < z < 1.629)$ | M1 | continuity corrections 20.5 or 21.5, 34.5 or 35.5

$= 0.8589 + 0.9483 - 1 = 0.807$ | M1 | $\Phi_1 + \Phi_2 - 1$
| A1 | answer rounding to 0.807
| [5] |
2 On average, 2 apples out of 15 are classified as being underweight. Find the probability that in a random sample of 200 apples, the number of apples which are underweight is more than 21 and less than 35.

\hfill \mbox{\textit{CAIE S1 2010 Q2 [5]}}
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