Answer
Yes, it would be unusual to see at least 260 adult Americans believe that the overall state of moral values in the US is poor. This is because $P(x \geq) = 0.0010$ is less than 0.05.
Work Step by Step
i) Verify that we can use the normal approximation to the binomial
$np(1-p) \geq 10$
= $500(0.45)(0.55) \geq$ 10
= $123.75 \geq 10$
Thus, we can use the normal distribution to approximate the binomial.
ii) Find mean and standard deviation of the data
$\mu = np = 500 \times 0.45 = 225$
$\sigma = \sqrt {np(1-p)} = \sqrt{123.75}$
iii) Want to find $P(x \geq 260)$
Apply the continuity correction: $P(x \geq 260)$ = $P(x \geq 259.5)$
iv) Find $P(x \geq 259.5)$
Convert 259.5 to a z score
z = $\frac{259.5-225}{\sqrt{123.75}}= 2.29$
Therefore, $P(x \geq 259.5)$= $P(z > 3.10)$ $= 1- 0.9990$ = 0.0010
