Answer
$P(x = 80) \approx 0.0995$
Work Step by Step
i) Verify that we can use the normal approximation to the binomial
$np(1-p) \geq 10$
= $100(0.8)(0.2) \geq$ 10
= $16 \geq 10$
Yes, we can use the normal distribution to approximate the binomial.
ii) Find mean and standard deviation of the data
$\mu = np = 100 \times 0.80 = 80$
$\sigma = \sqrt {np(1-p)} = \sqrt 16$ = 4
iii) Want to find $P(x = 80)$
Apply the continuity correction: $P(x = 80)$ = $P(79.5 \leq x \leq 80.5)$
iv) Find $P(79.5 \leq x \leq 80.5)$
Convert 79.5 and 80.5 to z-scores
z = $\frac{79.5-80}{4} = -0.125$
z = $\frac{80.5-80}{4} = 0.125$
Therefore, $P(79.5 \leq x \leq 80.5)$ = $P(-0.125 < z < 0.125)$
$= P(z < 0.125) - P(z < - 0.125)$
$= 0.54975 - 0.45025$
$= 0.0995$
