Markku M. answered • 02/05/20

PhD Candidate in Biostatistics who enjoys teaching statistics

In this case you are using the central limit theorem to approximate a normal distribution for a proportion.

So p = y/n, where y is the number of people who said yes and n is the number of people who said yes.

So p = 178/514 = .346

We want to check if p is > .3., where we say p0 = .3.

So our null hypothesis is that p <= .3 and our alternative hypothesis is p >.3

so to get our test statistics we do (p - p0)/sqrt((p0*(1-p0))/n) = (.346-.3)/sqrt((.3*(1-.3))/514)

so Z = .046/0.0202 = 2.28, so now we can calculate our p-value. the definition of our p-value is the probability of observing values like the values from the data or more extreme. So that would mean since our alternative hypothesis is p > .3, then we want P(Z>2.28) = 0.0113 = p-value.

Now we compare our p-value to our level of significance which is 0.05. Since our p-value = 0.0113 which is less than our alpha of 0.05 we reject the null hypothesis. This suggest that there is evidence to support the proportion of people who consider a career in science is greater than 30%.