Eat your cereal: A cereal manufacturer claims that the weight of a box of cereal labeled as weighing 12 ounces has a
a. If the claim is true, what is
b. Based on the answer to part (a), if the claim is true, is 11.99 ounces an unusually small mean weight for a sample of 75 boxes?
c. If the mean weight of the boxes were 11.99 ounces, would you be convinced that the claim was false? Explain.
d. If the claim is true, what is
e. Based on the answer to part (d), if the claim is true, is 11.97 ounces an unusually small mean weight for a sample of 75 boxes?
f. If the mean weight of the boxes were 11.97 ounces, would you be convinced that the claim was false? Explain.
a.
Find the probability
Answer to Problem 31E
The probability
Explanation of Solution
Calculation:
The given information is that the manufacturer claims that the weight of a box of cereal labelled as weighing 12 ounces has mean 12.0 ounces and standard deviation of 0.1 ounce. The sample of boxes is 75. Also,
Central limit theorem:
If
From the given information, it is observed that the simple size n is 75 and it is greater than 30. Therefore, it is appropriate to use the normal distribution.
For mean
From the given information, the mean of the population
For standard deviation
From the given information, the standard deviation of the population
The probability
Software Procedure:
Step by step procedure to find the probability by using MINITAB software is as follows:
- Choose Graph > Probability Distribution Plot > View Probability > OK.
- From Distribution, choose ‘Normal’ distribution.
- Enter Mean as 12.0 and Standard deviation as 0.0115.
- Click the Shaded Area tab.
- Choose X value and Left Tail for the region of the curve to shade.
- Enter the X value as 11.99.
- Click OK.
Output using MINITAB software is as follows:
From the output, it can be observed that the probability
b.
Check whether 11.99 ounces is an unusually small mean weight for a sample of 75 boxes based on the answer to part (a).
Answer to Problem 31E
No, the 11.99 ounces is not unusually small mean weight for a sample of 75 boxes based on the answer to part (a).
Explanation of Solution
Unusual:
If the probability of an event is less than 0.05 then the event is called unusual.
From part (a), the probability
Here, the probability
c.
Explain whether the person can be convinced that the claim is false if the mean weigh of the boxes were 11.99 ounces.
Answer to Problem 31E
No, the person cannot be convinced that the claim is false if the mean weight of the boxes were 11.99 ounces because the result is not unusual if the claim is true.
Explanation of Solution
From part (b), it can be observed that the 11.99 ounces is not unusually small mean weight for a sample of 75 boxes based on the answer to part (a). Therefore, the person cannot be convinced that the claim is false if the mean weigh of the boxes were 11.99 ounces because the result is not unusual if the claim is true.
d.
Find the probability
Answer to Problem 31E
The probability
Explanation of Solution
Calculation:
The probability
Software Procedure:
Step by step procedure to find the probability by using MINITAB software is as follows:
- Choose Graph > Probability Distribution Plot > View Probability > OK.
- From Distribution, choose ‘Normal’ distribution.
- Enter Mean as 12.0 and Standard deviation as 0.0115.
- Click the Shaded Area tab.
- Choose X value and Left Tail for the region of the curve to shade.
- Enter the X value as 11.97.
- Click OK.
Output using MINITAB software is as follows:
From the output, it can be observed that the probability
e.
Check whether 11.97 ounces is an unusually small mean weight for a sample of 75 boxes based on the answer to part (d).
Answer to Problem 31E
Yes, the 11.97 ounces is unusually small mean weight for a sample of 75 boxes based on the answer to part (d).
Explanation of Solution
Calculation:
From part (d), the probability
Here, the probability
f.
Explain whether the person can be convinced that the claim is false if the mean weight of the boxes were 11.97 ounces.
Answer to Problem 31E
Yes, the person can be convinced that the claim is false if the mean weigh of the boxes were 11.97 ounces because the result is unusual if the claim is true.
Explanation of Solution
From part (e), it can be observed that the 11.97 ounces is unusually small mean weight for a sample of 75 boxes based on the answer to part (d). Therefore, the person can be convinced that the claim is false if the mean weigh of the boxes were 11.97 ounces because the result is unusual if the claim is true.
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