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amm1812
3 months ago
7

Use the normal approximation to the binomial distribution to answer this question. Fifteen percent of all students at a large un

iversity are absent on Mondays. If a random sample of 12 names is called on a Monday, what is the probability that four students are absent?
Mathematics
1 answer:
AnnZ [12.3K]3 months ago
3 0

Answer: 0.1289

Step-by-step explanation:

Given: The proportion of students absent on Mondays at a large university.: p=0.15

Sample size: n=12

Mean: \mu=np=12\times0.15=1.8

Standard deviation = \sigma=\sqrt{np(1-p)}

\Rightarrow\ \sigma=\sqrt{12(0.15)(1-0.15)}=1.23693168769\approx1.2369

Let x represent a binomial variable.

Referencing the standard normal distribution table,

P(x=4)=P(x\leq4)-P(x\leq3) (1)

Z score for normal distribution:-

z=\dfrac{x-\mu}{\sigma}

For x=4

z=\dfrac{4-1.8}{1.2369}\approx1.78

For x=3

z=\dfrac{3-1.8}{1.2369}\approx0.97

Thus, from (1)

P(x=4)=P(z\leq1.78)-P(z\leq0.97)\\\\=0.962462-0.8339768\approx0.1289

Consequently, the likelihood of four students being absent = 0.1289

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