Hypothesis test with given variation

I am trying to solve the following problem. But I don't know what is the appropriate formulas I have to use.

Could you help me please ?

A new machine is tested in a factory. It is said that the average weight of a generated product is at least 60kg and variation 3.2kg. The director of the factory worries that if the average weight is less than 60kg the factory will have problem. A sample of 45 products have a average 58.1kg a. In significance value 0.05, is the average weight less than 60kg ? b. If the variation is 4.2 kg what is the answer in a?

By hypothesis, $X$=weight of a product has mean $60$ and standard deviation $3.2$. $Y$=the average weight of $45$ products has mean $60$ and standard deviation $\frac{3.2}{\sqrt{45}}=0.477$
To get the critical value, we calculate $60-0.477\times 1.64=59.22$ The observed value is smaller, so we can concolude at the $5$%-level that the average weight is less than $60$.
If the standard derivation is $4.2$, we get a critical value $58.97$, so the conclusion is the same.