A spinach producer is testing a new packaging line They desi

A spinach producer is testing a new packaging line. They desire that the mean weight of spinach in each package is 8 ounces. If the mean is too high, they will lose money, and if the mean is too low, customers will get angry. They run the machine for a few days to get a large population of packages, then select 12 packages at random and weigh the spinach in each selected package. If they find evidence that the mean is too high or too low, they can recalibrate the machine and try again. The results are below (in ounces):

7.7, 6.8, 8.0, 7.4, 7.1, 7.4, 7.2, 7.3, 8.3, 7.7, 7.6, 7.0

(c) Choose an appropriate test statistic for this situation and justify your answer. Then compute the observed value of the test statistic for this data.

(d) Find the rejection region if we desire a test with = 0.01.

(e) Make a reject or not reject conclusion. Then state the conclusion in the context of the problem. In other words, does it seem the packaging line needs recalibrating, and if so, in which direction?

thank you!^^

Solution

Formulating the null and alternative hypotheses,              
              
Ho:   u   =   8  
Ha:    u   =/   8  
              
As we can see, this is a    two   tailed test.      

c)

As n = 12 < 30, we use the t statistic.

Getting the test statistic, as              
              
X = sample mean =    7.458333333          
uo = hypothesized mean =    8          
n = sample size =    12          
s = standard deviation =    0.427377486          
              
Thus, t = (X - uo) * sqrt(n) / s =    -4.390470809   [ANSWER, TEST STATISTIC]

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d)

Thus, getting the critical t,              
df = n - 1 =    11          
tcrit =    +/-   3.105806516      
              
Thus, we reject Ho if |t| > 3.1058. [ANSWER, REJECTION REGION]

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e)

As |t| > 3.1058, we   REJECT THE NULL HYPOTHESIS.

Thus, there is significant evidence at 0.01 level that the mean weight of each package is not 8 oz.

As t < 0, then it needs to calibrate so that the mean will be higher (towards the positive direction). [ANSWER]