Hypothesis finding type 1 error probability The manufacturer of bags of cement claims that they fill each bag with at least 50.1
pounds of cement. Assume that the standard deviation for the amount in each bag is
1.2 pounds. The decision rule is adopted to shut down the filling machine if the
sample mean weight for a sample of 40 bags is below 49.7. What is the probability of
a Type I error?
Please help 
 A: Z Scores and Normal Distributions
Because you know the population standard deviation and your sample is over 30, you can use a z-test to answer this question. I'm assuming this is in the context of normally distributed cement bags. The first thing you need to do is convert the "cutoff" value of 49.7 into a z-score:
Calculate the z-score
Here's what we know:
\[\mu = 50.1\]
\[X = 49.7\]
\[\sigma = 1.2\]
\[n = 40\]
Here's the formula for a z-score:
\[ z = {\mu-X\over {\sigma\over \sqrt{(n)}}} \]
Plug in the numbers:
\[ z = {49.7-50.1\over {1.2\over \sqrt{(40)}}}  = -2.108\]
Use Z-score to find tail probability (AKA type I error)
Great! So now the z-value of our cutoff metric is -2.108. 
You can then use a z-table (I found one here) to calculate the tail probability for that z-value. Here, tail probability is the same as your Type I error. 
To use the Z-table, look up the relevant row (to the tenth decimal) and match with the relevant column (to the hundredth decimal). I've highlighted the correct row x column for you convenience. 
Note that I rounded your z of -2.108 to -2.11. 

If you want a more accurate answer, I used python's scipy module to calculate the exact tail probability at 0.0175. Pretty close to the z-table's answer, actually. 
