Did I use z-test correctly in this problem?

The problem:

It has been said that it is difficult to distinguish Coke and Pepsi by taste alone, without the visual cue of the bottle or can. In an experiment that I did in a class at Stanford, 10 cups were filled at random with either Coke or Pepsi. A student volunteer tasted each of the 10 cups and correctly named the contents of seven. Is this sufficient evidence to conclude that the student can tell apart Coke and Pepsi?

According to the question, we define the null and Alternative hypotheses:

• Null hypothesis (H0): The student cannot distinguish between Coke and Pepsi by taste alone.
• Alternative hypothesis (Ha): The student can distinguish between Coke and Pepsi by taste alone.

According to the z-table, the p-value for a z-value of 4 is approximately 0.00001. Since 0.00001 is smaller than 0.05, we reject the null hypothesis. We are convinced that the student can distinguish Coke and Pepsi by taste.

I guess I have followed the process correctly, but the z-value and p-value seem a bit strange! 0.00001 is very small value!

• In HW problems, the math often works out neatly. I don't see any errors. As for your p value, you might want to check the table, or use an online calculator, or R or whatever. Jun 18, 2023 at 18:42
• You might be interested in a problem called “the lady tasting tea”.
– Dave
Jun 19, 2023 at 3:43

This is a common trap when calculating the $$z$$-transform of a binomial random variable that represents a Bernoulli experiment: confounding the number $$x$$ of positive outcomes with the mean $$\bar x$$, which in the present example would be $$\bar x = 7/10$$, not $$7$$.
For the $$z$$-transform, divide by the SE of the statistic you used, which in your case is $$\sigma$$, not $$\sigma/\sqrt{n}$$, since you denoted by $$\bar x$$ the number of positive outcomes.