Part a)
p = population proportion of correct guesses
Null Hypothesis: p = 0.5
Alternative Hypothesis: p > 0.5
The claim that people can tell the difference of the two drinks is in the alternative hypothesis p > 0.5 meaning that it's more than just luck at play here. Saying p = 0.5 is basically saying there's a coin toss to determine the guess.
x = 53, n = 80
phat = x/n = 53/80 = 0.6625
SE = sqrt(p*(1-p)/n) = sqrt(0.5*(1-0.5)/83) = 0.05488212999484
z = (phat-p)/(SE)
z = (0.6625-0.5)/(0.05488212999484)
z = 2.96089091322218
z = 2.96
Use a table or calculator to find that
P(Z < 2.96) = 0.9985
So,
P(Z > 2.96) = 1-P(Z < 2.96)
P(Z > 2.96) = 1-0.9985
P(Z > 2.96) = 0.0015
The p-value is approximately 0.0015. This value is less than many alpha values commonly used (such as 0.01 or 0.05) so we reject the null hyptohesis that p = 0.5 and accept that p > 0.5 is true. So the participants aren't randomly guessing. The results are significantly better than random guesses.
------------------------------------------------------------------------------------------------
Part b)
The p value 0.0015 is the probability of observing 53 or more claims out of 80 claims, under the assumption that the null hypothesis is true. This p value is very very small, so there is a small chance that the null hypothesis is correct. The smaller the p value, the more likely we reject the null. The general rule is that if the p value is smaller than the significance level alpha, we reject the null. Your textbook will state what alpha is equal to. If not, then the default is alpha = 0.05
