Question 1182532: Cough-a-Lot children’s cough syrup is supposed to contain 6 ounces of medicine per bottle. However, since the filling machine is not always precise, there can be variation from bottle to bottle. The amounts in the bottles are normally distributed with σ = 0.3 ounces. A quality assurance inspector measures 10 bottles and finds the following (in ounces):
5.95
6.10
5.98
6.01
6.25
5.85
5.91
6.05
5.88
5.91
Are the results enough evidence to conclude that the bottles are not filled adequately at the labeled amount of 6 ounces per bottle?
State the hypothesis you will test. (2 pts)
Calculate the test statistic. (2 pts)
Find the P-value. (2 pts)
What is the conclusion? (2 pts)
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Add up the data values to get
5.95+6.10+5.98+6.01+6.25+5.85+5.91+6.05+5.88+5.91 = 59.89
Then divide that sum over n = 10
xbar = (sum of values)/n
xbar = (59.89)/10
xbar = 5.989
this is the sample mean
We're given sigma = 0.3 which is the population standard deviation. Because we know this value, we'll use the standard normal Z distribution.
Hypothesis
H0: mu = 6
H1: 
So either mu is 6 (null) or it's not 6 (alternative). The claim is in the null.
The alternative hypothesis tells us we have a two tailed test.
Test statistic:
z = (xbar - mu)/(sigma/sqrt(n))
z = (5.989 - 6)/(0.3/sqrt(10))
z = -0.11595018087283
z = -0.12
Now you have a few options at this point to find the p-value. One path is to use a table of values. Another is to use a graphing calculator or computer software.
If you go with the software option, then you can use something like this
https://www.davidmlane.com/hyperstat/z_table.html
Leave the "Area from a value" radio button as it is. Also, leave the "Mean" and "SD" boxes as 0 and 1 respectively.
Then click on the "below" radio button. In the box next to it, type in -0.12 and hit "recalculate"
The result 0.4522 should pop up
This represents P(z < -0.12) = 0.4522 approximately.
It's the approximate area under the Z curve that's to the left of -0.12
Double this value to get 2*0.4522 = 0.9044
This is the approximate p-value for the two tailed test.
This is a very large p-value and it means we effectively fail to reject the null regardless of what alpha is.
The rule is:
If the p-value is smaller than alpha, then you reject the null.
One way to remember is the phrase "if the p-value is low, then the null must go"
Conclusion:
We failed to reject the null, so we must conclude that mu = 6. Therefore, the average bottle contains 6 ounces of medicine.
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