What's the probability that the news station predicted the outcome incorrectly?

Question

I wrote this problem for a high school statistics activity on the sampling distribution of sample proportions. When I sat down to write solutions I realized that it was maybe too confusing to assign, and now I'm not confident in my solution. Is my attempt a reasonable solution?

Question:

Suppose that during an election, a news agency that conducts a poll of 105 random voters and finds a sample proportion of 54% in support of a ballot measure that requires a majority to pass. The agency publishes a prediction that the measure will pass. How likely is it that the news agency is incorrect?

Attempt at a solution:

Let $n = 105,$ $\hat p=0.54$, and we use $ \sigma = \sqrt{\dfrac{0.25}{105}} = 0.0488$ as the worst case, since $p$ is unknown.

Notice that 0.54 is $0.4 = 0.82 \sigma$ away from $0.50$, which is the threshold for a “majority”. The proportion of samples with sample proportions that have z-scores within $.82$ of $0$ is $normalcdf(-.82,.82,0,1) = 0.587$.

So, 58.7% of random samples will have sample proportions within $.82\sigma$ of the true population proportion. In order for the news agency to be incorrect, $ \hat p=0.54$ must be more than $0.82 \sigma$ above the true population proportion.

Because of the symmetry of the normal distribution, at most $(1 - 0.587)/2 = 20.65\%$ of random samples will be at least this far above true proportion. So, since the sample was random, there is at most a $20.65\%$ chance that the outcome of the election will disagree with the news agency’s prediction.

Answer 1

Printout from a recent release of Minitab statistical software. I entered 57 Yes's out of the 105 interviews, this gives the estimate $\hat p = 542857,$ not $\hat p = 0.54.$ Why not give the actual number in favor?

Test and CI for One Proportion 

Test of p = 0.5 vs p > 0.5


                                              Exact
Sample   X    N  Sample p  95% Lower Bound  P-Value
1       57  105  0.542857         0.457999    0.218

P-value exceeds 5% = 0.05, so you cannot reject $H_0.$ The actual percentage of the population in favor might be as low as 45.8%.

With so small a sample, it is not surprising that the poll does not give useful results. In election polling, it takes about $n=600$ subjects to get minimally useful results. This sample size gives a 2-sided confidence interval with a margin of sampling error about $\pm 4\%.$ (For $\pm 2\%$ use $n = 2500.)$