Hypothesis Testing

#1
Hi guys!
I have a statistics question that I am completely stumped with:

From a sample of people (n=36), with certain eating habits, a mean cholesterol value of 4.1 mmol/l was calculated. From a medical reference table, it is shown that 95% of all values are between 3.2 – 6.2 mmol/l for the actual age group of the patients (tip: 95% of the values are between µ±1.96σ). The cholesterol measurements follow a normal distribution. Can we conclude that the sample group have a different cholesterol value than “the normal population”? Show all the steps to solve this problem and use α = 0.05.

Here is what I have done:
H0: µ = 4.7
HA: = µ > 4.7

µ = 4.1
x = 4.7 (6.2-3.2/2)
α = 0.05 = a critical value of Z= 1.645
n = 36s = 1.96
df = t0.025,35

Using a z-score formula ( =−/√)
=4.7 − 4.11.96/√36
z = 1.836
As the calculated z-score is 1.836, it exceeds the α=0.05 critical value of 1.645. Subsequently, this means H0 can be rejected and our alternative hypothesis accepted. We can conclude that the sample group does in fact have a different cholesterol value than the “normal population”.

Though this was incorrect. Can anyone help me understand the correct working out?

Thank you!