The method I'm going to use here relies on the classical frequentist model of statistics. There are other approaches, which, believe it or not, give other answers, but the classical frequentist model is the most widely used.
The method is as follows:
- Assume for the moment that exactly 30% of the population have considered careers in science.
- Ask "if the assumption in part 1 were true, what is the probability that at least 178 out of 514 people would say they were interested in a career in science?" Calculate that probability and call it p.
- If p is less than our α level, (in this case α=0.05), then we conclude that more than 30% of the population is interested in a career in science.
The only hard part of the calculation is part 2). If exactly 30% of the population were interested in a career in science, what would happen if we asked 514 people? We don't know exactly, because there's some randomness depending on which people we happened to ask. However, we can calculate the mean and standard deviation of the number of "yes" answers.
The mean can be calculated
μ=0.30*514 = 154.2
We don't care that the mean is not a whole number, this is perfectly fine. The standard deviation can be calculated
σ = sqrt(0.3*0.7*514) = 10.39
Next, we ask "what is the probability that we'd get at least 178 "yes" answers?" We want to use the normal distribution. The one hitch is that the normal distribution can take any number, but the number of people who answer "yes" must be a whole number, so we use rounding for the normal distribution. Thus, to get at least 178 "yes" answers, we need the normal distribution to give us an answer of at least 177.5.
For a normal distribution with mean 154.2 and standard deviation 10.39 to give us a value of 177.5, we need a z-score of
z = (177.5 - 154.2)/10.39 = 2.24
Using a normal distribution table, we find that a z-score this high will only happen around 1% of the time. Because 1% is less than our α level of 5%, we conclude that it is likely that the true proportion of people interested in a career in science is more than 30%.