Coffee sales - stats question relating to p-value and test statistics.

Question

I need help mainly with part c of question 11.

My work is below.

Answer from the back of the book:

Answer 1

Z-test of $H_0: \mu = 354$ against $H_a: \mu > 354$, based on $\sigma = 33$, and $n = 3$ observations with mean $\bar X = 398$. Then $Z = (398 - 354)\sqrt{3}/33 = 2.3094$.

The P-value is the probability of a result as or more extreme (here larger) than the observed test statistic 2.3094. This is the area under the standard normal curve in the right-hand tail beyond 2.3094.

From software this area (probability) is 0.01046, rounded to five places. Using a printed table of the normal distribution you will likely need to round the value of the test statistic to 2.31 and read a four-place result. Consistent with this, the answer book has 0.0104.

We reject $H_0$ at the 5% level of significance because 0.0104 < .05, but we do not reject at the 1% level because 0.0104 > .01.

Note: you were asked to sketch the area under the distribution curve for $\bar x$. This is different than the standard normal curve that applies to the test statistic Z. Because each of the three observations $X_i \sim Norm(354, 33^2)$, it follows that $\bar X \sim Norm(354, 33^2/3)$. To shade the area under this curve that corresponds to the P-value, you will have to solve $(\bar X - 354)\sqrt{3}/33 = 2.31$ to get a bound for $\bar X$. According to the somewhat smudgy shot of your work page, it seems to me you may have sketched the wrong distribution and shaded something in the wrong tail.