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Feb/Mar 2021 p52 q7
3285
On average at all the schools in this country 30% of the students do not like any sports.
(ii) 90 students from this country are now chosen at random.
Use an approximation to find the probability that fewer than 32 of them do not like any sports.
Solution
Let the random variable \(X\) represent the number of students who do not like any sports. \(X\) follows a binomial distribution with parameters \(n = 90\) and \(p = 0.3\).
To approximate, we use a normal distribution with mean \(\mu = np = 0.3 \times 90 = 27\) and variance \(\sigma^2 = np(1-p) = 0.3 \times 90 \times 0.7 = 18.9\).
Thus, \(X \sim N(27, 18.9)\).
We want \(P(X < 32)\). Using continuity correction, this is approximated by \(P(X < 31.5)\).
