SOLUTION: Each bottle of fruit juice from a small manufacturing plant is supposed to contain exactly 12 fluid ounces of juice. Susan is in charge of quality control and decided to tes

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Question 1010759:
Each bottle of fruit juice from a small manufacturing plant is supposed to contain exactly 12 fluid ounces of juice. Susan is in charge of quality control and decided to test this claim by gathering a SRS of 30 bottles. She will recalibrate the machinery if the average amount of juice per bottle differs from 12 fluid ounces at the 1% level of significance. Her sample of 30 bottles has an average of 11.92 fluid ounces and a sample standard deviation of 0.26 fluid ounces. Conduct a hypothesis test to determine if the machinery needs recalibrated.
Found 2 solutions by Theo, stanbon:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!


since the standard deviation comes from the sample, you would conduct a t-test.

the degrees of freedom is equal to n-1 which is equal to 30 - 1 which is equal to 29.

the critical t-score at .01 alpha with 29 degrees of freedom is plus or minus 2.46202 truncated to 5 decimal places.

the sample standard deviation is .26 ounces.

the standard error of the distribution of sample means is .26 / sqrt(30) = .04746 truncated to 5 decimal places.

you can calculate the critical raw scores from the critical t-scores.

the critical t-scores at .01 significance level and 29 degrees of freedom at each end are plus or minus 2.46202 truncated at 5 decimal places.

the formula is:

t = (x-m)/s

t is the t-score.
x is the mean of the sample.
m is the desired measurement of 12.
s is the standard error.

from this formula, solve for x to get:

x = t*s+m

when t = 2.46202, this becomes x = 2.46202 * .04746 + 12 = 12.11684 truncated to 5 decimal places.

when t = -2.46202, this becomes x = -2.46202 * .04746 + 12 = 11.88315 truncated to 5 decimal places.

the mean of the sample is 11.92 which is within the critical levels of 11.88 to 12.12 rounded to 2 decimal places.

no calibration is required.


Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Each bottle of fruit juice from a small manufacturing plant is supposed to contain exactly 12 fluid ounces of juice. Susan is in charge of quality control and decided to test this claim by gathering a SRS of 30 bottles. She will recalibrate the machinery if the average amount of juice per bottle differs from 12 fluid ounces at the 1% level of significance. Her sample of 30 bottles has an average of 11.92 fluid ounces and a sample standard deviation of 0.26 fluid ounces. Conduct a hypothesis test to determine if the machinery needs recalibrated.
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Ho: u = 12
Ha: u # 12
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z(11.92) = (11.92-12)/[0.26/sqrt(30)] = -1.6853
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p-value = 2*P(z < -1.6853) = 2*0.0460 = 0.0920
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Since the p-value is greater than 1% fail to reject Ho.
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Conclusion:: Based on these results DO NOT recalibrate the machine.
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Cheers,
Stan H.
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