- Sample mean can be calculated using the following formula:
μ=x=1 Nxi ,
where xi is mass of sample.
The mean is equal
x=130i=130xi=14.87 ounces
In a set of observations, median is a value of variable that have half of the number of observations below it and remaining half above it (Agarval, 43). To set median it is necessary to arrange data in ascending or descending order. As the number of samples is even (N=2p) the average of p-th and (p+1)-th is median. In our case median is equal to 14.8 ounces.
Standard deviation can be calculated as (Bajpai, 129)
σ=i=1Nxi-μ2N=0.541 ounces
σN=0.54130=0.099 ounces
It is possible to find numbers –x and x, between which lies mass of soda with probability 95 % (1-α).
Therefore:
P-x≤X≤x=1-α=0.95
The cumulative normal distribution function is
Φx=PX≤x=1-α2=0.975
x=Φ-1Φx=1.96 (Bajpaj, 313)
We get
0.95=1-α=P(x-1.96σN≤μ≤x+1.96σN)
Lower endpoint is 14.87-1.96*0.099=14.68 ounces
Upper end point is 14.87+1.96*0.099=15.06 ounces
It means that in 95% of the cases µ will be in the calculated Confidence Interval (between 14.68 and 15.06 ounces).
Our null hypothesis is that µ=16 ounces. The alternative hypothesis is that µ<16. For 95 % Confidence Level z=±1.645. We calculate z on the next step.
z=x-μσN=14.87-160.54130=-1.130.099=-11.44
The calculated z value is less than -1.645. It means that null hypothesis should be rejected and alternative hypothesis (µ<16) should be accepted. Besides 16 ounces does not belong to the Confidence Interval.
- Test results show that there are less than 16 ounces of soda in each bottle. The most probable cause is that automat for filling bottles is not well calibrated. It gives less soda than it is necessary. The second reason can be the changes of bottle mass. To find the soda mass you may weigh bottle with soda subtracting from this mass the mass of bottle. If this procedure is used and bottle mass had changed, the results of calculation can be less than it is expected. It is also possible that if you pour soda from bottle to measuring vessel some amount of soda is left in the bottle. It is also possible that volume of bottle changed.
References
Agarval, B.L. Basic Statistics; New Age International (P) Ltd, Publishers: New Delhi, 2006, pp 43.
Bajpai, Naval Business Statistics; Dorling Kindersley (India) Pvt. Ltd.: New Delhi, 2010, pp 311-320.
