Federal law requires that a jar of peanut butter that is labeled as containing 3

Question

Federal law requires that a jar of peanut butter that is labeled as containing 32 ounces must contain at least 32 ounces. A consumer protection organization thinks that the manufacturer of one brand of peanut butter is shortchanging customers by intentionally underfilling the jars. Technicians in the organization's testing lab carefully weigh a random sample of 25 jars of the peanut butter, empty the jars completely, clean and dry them, and then reweigh the empty jars. The weight of each empty jar is subtracted from the first weight. The mean weight of the peanut butter in the sample is 31.48 ounces, and the sample standard deviation is 1.056 ounces. Results of exploratory data analyses indicate that the sample was drawn from a normal distribution. The null hypothesis is tested at the significance level a = 0.01. Conduct a one-sample f test to determine whether there is sufficient evidence to reject the null hypothesis. Should you do a one-tailed test or a two-tailed test? Interpret the meaning of the results in the context of this study. If the null hypothesis had been tested at the .05 level instead of at the .01 level, would you reach the same conclusion? Explain. Using the procedure from section 9.3, construct a 98% confidence interval for the population mean weight of peanut butter. Compare the confidence interval to the results of the f test you conducted in part (a).

Explanation / Answer

a)

As we claim underfilling, we do a one tailed test. [ANSWER, ONE TAILED]

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Formulating the null and alternative hypotheses,              
              
Ho:   u   >=   32  
Ha:    u   <   32  
              
As we can see, this is a    left   tailed test.      
              
df = n - 1 =    24          
              
Getting the test statistic, as              
              
X = sample mean =    31.48          
uo = hypothesized mean =    32          
n = sample size =    25          
s = standard deviation =    1.056          
              
Thus, t = (X - uo) * sqrt(n) / s =    -2.462121212          
              
Also, the p value is              
              
p =    0.010688895          
              
As P > 0.01, we   FAIL TO REJECT THE NULL HYPOTHESIS. [ANSWER]          
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b)

Hence, there is no significant evidence of underfilling of peanut butter jars by the manufacturer at 0.01 level. [CONCLUSION]

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c)

NO. In that case, P < 0.05, and we reject the null hypothesis.

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d)

Note that              
Margin of Error E = t(alpha/2) * s / sqrt(n)              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.01          
X = sample mean =    31.48          
t(alpha/2) = critical t for the confidence interval =    2.492159473          
s = sample standard deviation =    1.056          
n = sample size =    25          
df = n - 1 =    24          
Thus,              
Margin of Error E =    0.526344081          
Lower bound =    30.95365592          
Upper bound =    32.00634408          
              
Thus, the confidence interval is              
              
(   30.95365592   ,   32.00634408   ) [ANSWER]

As we can see, 32 is inside this interval, so it is consistent with part a) that there is no significant evidence that the true mean is less than 32.

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