Question

In a city, 51 % of the population favors the incumbent, Dawn Morgan, for mayor. A simple random sample of 450 is taken. Use the normal approximation along with the continuity correction factor to find the probability that at most 230 respondents favor Dawn Morgan for mayor.

• Let XX be the number of respondents out of 450 that favor Dawn Morgan for mayor. Describe the distribution of XX and its parameters:

~  Select an answer χ² F T N B  ( n=n= , p=p= )

• Use the random variable notation to symbolically express the probability that at most 230 respondents favor Dawn Morgan for mayor:

Select an answer P(X≤230) P(X>230) P(X<230) P(X≥230)

• Let YY be a normal variable that will be used to approximate the probability in question. Find the parameters of YY (round the answers to 2 decimal places):

Y~Y~ Select an answer B F T N χ²  ( μ=μ= , σ=σ= )

• Use the random variable notation to symbolically express the approximate probability that at most 230 respondents favor Dawn Morgan for mayor:

Select an answer P(Y>230) P(Y<230)

• Use the correction for continuity:

Select an answer P(Y>229.5) P(Y<230.5) P(Y>230.5) P(Y<229.5)

• Find the probability (round the answer to 4 decimal places):

Answer 1

X follows Binomial (n = 450, p = 0.51 )

The probability that at most 230 respondents favor Dawn Morgan for mayor = P(X≤230)

If normal distribution is used to approximate above probability then

μ = np = 450 * 0.51 = 229.5

Hence

Y ~ N ( μ = 229.5 , σ = 10.60 )

Approximate notation = P(Y<230)

Continuity correction = P(Y<230.5)