Question

An opinion poll was taken before an election. The table shows the percentage of voters who said they would vote for parties $A, B$ and $C$ .

Party	$A$	$B$	$C$
Votes $(%)$	$36$	$41$	$23$

Find an approximation for the probability that, in a random sample of $120$ of these voters exactly $50$ said they would vote for party $B$ .

Accepted Answer

Given that table of opinion poll, which was taken before election,

Party $A$ $B$ $C$

Votes $(%)$ $36$ $41$ $23$

Taking a random sample of $120$ of these voters.

So, $n = 120$

Let random variable $X$ is the number of voters, who said they would vote for party $B$ .

Let success is the voter said they would vote for party $B$ .

So, probability of success $p = 41 % = 0.41$

We know that for the binomial distribution $X ~ B (n, p)$ , $n$ is a number of trials and $p$ is the probability of success.

Here $X ~ B (120, 0.41)$

Probability of failure is $q = 1 - p = 1 - 0.41 = 0.59$

A binomial distribution can be well-approximated by a normal distribution if $n p$ and $n q$ are both greater than $5$ .

Here, $n p = 120 \times 0.41 = 49.2 > 5$ ,

$n q = 120 \times 0.59 = 70.8 > 5$

Hence, given binomial distribution can be well-approximated by a normal distribution.

Mean $μ = n p = 49.2$ ,

Variance $σ^{2} = n p q = 120 \times 0.41 \times 0.59 = 29.028$

Probability that exactly $50$ voters said they would vote for party $B$ is $P (X = 50)$ .

$P (X = 50) = P (49.5 < X < 50.5) \Rightarrow P (X = 50) = P (X < 50.5) - P (X < 49.5)$

Using the continuity correction, standardising the value of $x = 50.5$ and $x = 49.5$ .

Standardising value $z = \frac{x - μ}{σ}$

Here, $μ = 49.2, σ^{2} = 29.028$

For $x = 50.5$ ,

$\Rightarrow z = \frac{50.5 - 49.2}{\sqrt{29.028}} \Rightarrow z = \frac{1.3}{5.388} = 0.241$

For $x = 49.5$ ,

$\Rightarrow z = \frac{49.5 - 49.2}{\sqrt{29.028}} \Rightarrow z = \frac{0.3}{5.388} = 0.056$

So, $P (X = 50) = P (Z < 0.241) - P (Z < 0.056)$

We know that, if $z = a, P (Z < a) = ϕ (a)$

$\Rightarrow P (X = 50) = ϕ (0.241) - ϕ (0.056) \Rightarrow P (X = 50) = 0.5952 - 0.5223 \Rightarrow P (X = 50) = 0.0729$

Dean Chalmers and Julian Gilbey Solutions for Chapter: The Normal Distribution, Exercise 11: EXERCISE 8E

Dean Chalmers Mathematics Solutions for Exercise - Dean Chalmers and Julian Gilbey Solutions for Chapter: The Normal Distribution, Exercise 11: EXERCISE 8E

Questions from Dean Chalmers and Julian Gilbey Solutions for Chapter: The Normal Distribution, Exercise 11: EXERCISE 8E with Hints & Solutions

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