The confidence that the box you bought did not come from the factory is 90 %.
Step-by-step explanation:
Since we have given that
Population Mean weight (\mu)= 10 grams a piece
Standard deviation of the weight of a box = 3 grams
Number of mints = 10
We need to buy a box of mints that weighs 95 grams.
Sample mean is given by
x = [tex]\frac{95}{10}=9.5[/tex] grams
.
First we find out the standard error which is given by
[tex]s=\frac{\sigma}{\sqrt{n}}\\\\=\frac{3}{\sqrt{10}}\\\\[/tex]= 0.94868
Since it is normal distribution, so, we will find z-score.
[tex]z=\frac{x-\mu}{s}\\\\z=\frac{9.5-10}{0.94868}\\\\z=-0.527\\\\[/tex]
z = - 0.53
The area to the left of a z-score of -0.53 = 0.29805.
So, it may be 90% or 95 % confidence.
For 95% confidence level,
[tex]\alpha=\frac{1-0.95}{2}=0.025[/tex]
Similarly,
For 90% confidence level,
[tex]\alpha=\frac{1-0.90}{2}=0.05[/tex]
The value is much smaller than 0.05.
So, we will get 90% confidence and the critical value = 1.645
Margin of error is given by
(Standard deviation) [tex]\times[/tex] (critical\ value) = 0.94868 [tex]\times[/tex] 1.645
=1.56
So, confidence interval will be
(10-1.56,10+1.56)
=(8.44,11.56)
