Stats confusion, testing a normal distribution

My textbook ‘explains’ this topic here:



And then here is a worked example:



I don’t understand why you cannot just use the fact that the sample mean is distributed by N(population mean, population variance/n) where n is the size of the sample, so using this you can work out the p-values for any obtained sample mean and compare this to the significance level.

For example, in the worked example I would’ve thought you could say:
1)Assume the null hypothesis is true so the mean is 30.0.
2)Therefore the sample mean is ~N(30.0, (0.16/32)) for a sample size of 32.
3)The p-value, i.e. the probability of getting a sample mean of 29.9 or less within this distribution is = 2.75*10^(-89) (from my calculator). This is (much) less than the significance level of 10% so it is ‘too unlikely to happen by chance’. Therefore our assumption that the mean is 30 is likely to be incorrect.

Instead of what I did, the method they follow is to calculate the corresponding z-value for this observed sample mean, and then calculate the p-value from that z-value in the standard distribution. I would’ve thought my method would give the exact same p-value, could someone please explain why my method is wrong?

The new standard deviation is ~0.07. So a difference in means of 0.1 is about 1.5 standard deviations. Testing at 10% is in line with these values.
What did you enter for your calculstor?
A value of 10^(-89) means you've entered the wrong thing or are calculating the wrong thing.