Answer
The probability that 164 voters voted for the referendum is 0.082, that is, approximately 9 voters in 100 samples of 310 support the cause.
The exit polling can be biased as the voter who voted against this cause will not want to admit it.
Work Step by Step
Here, a sample of 164 voters voted for the referendum. The sample size is 310 voters. The population proportion of voters that voted for the referendum is provided as 0.49. The mean is calculated as
\[\begin{align}
& \mu =p \\
& =0.49
\end{align}\]
and the standard deviation is calculated as
\[\begin{align}
& \sigma =\sqrt{\frac{p\left( 1-p \right)}{n}} \\
& =\sqrt{\frac{0.49\left( 1-0.49 \right)}{310}} \\
& =0.028
\end{align}\]
The sample proportion is calculated as
\[\begin{align}
& \hat{p}=\frac{x}{n} \\
& =\frac{164}{310} \\
& =0.529
\end{align}\]
The probability is calculated as
\[\begin{align}
& P\left( \hat{p}\ge 0.529 \right)=P\left( \frac{\hat{p}-\mu }{\sigma }\ge \frac{0.529-0.49}{0.028} \right) \\
& =P\left( Z\ge \frac{0.529-0.49}{0.028} \right) \\
& =P\left( Z\ge 1.39 \right)
\end{align}\]
The probability is calculated in Excel by using the formula\[=NORMSDIST\left( {} \right)\]
\[\begin{align}
& =P\left( Z\ge 1.39 \right) \\
& =1-P\left( Z<1.39 \right) \\
& =1-0.918 \\
& =0.082 \\
\end{align}\]
0.198
The voters that really want improvement in education by increased funding are 49%. So, approximately 9 voters result in 52% or more supporting this cause.
