Finding Probability of Low Cereal Content in Random Selection of Boxes

In summary, the conversation discusses a problem involving a machine filling cereal boxes with an assumed normal distribution of mean 22 oz for a supposedly 20 oz box. The question asks for the probability of finding an average content of less than 18 oz in a sample of four randomly selected boxes. The solution involves calculating the Z-score and using Table E to find the area, which results in a probability of 0.001. The confusion over the mean is clarified as being 22 oz.
  • #1
shawnz1102
26
0

Homework Statement


A machine fills cereal boxes at a factory. Due to an accumulation of small errors (different flakes sizes, etc.) it is thought that the amount of cereal in a box is normally distributed with mean 22 oz. for a supposedly 20 oz. box. Suppose the standard deviation of the amount filled is 1.3 oz. What is the probability that a federal regulatory selects four of these boxes at random and finds that the average content of these boxes is less than 18 oz?

The Attempt at a Solution


[PLAIN]http://img508.imageshack.us/img508/3904/49450328.jpg

From the graph, I calculated the Z score using the Z score equation for distribution of sample means.

The Z-score I got was -6.15

I then found the area from Table E which came out to be 0.4999 (Table E said any value greater than 3.09 use 0.4999. Since it's to the left of the mean, I subtracted 0.5 from the Z value (0.5-0.4999) and my answer came out to be 0.001

Can anyone please double check if I got the answer correct?

I'm a bit confused by the question when it says that it's normally distributed with a mean of 22 oz for a "supposedly 20 oz box." So is it implying that the mean is 22 or 20?
 
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  • #2
shawnz1102 said:

Homework Statement


A machine fills cereal boxes at a factory. Due to an accumulation of small errors (different flakes sizes, etc.) it is thought that the amount of cereal in a box is normally distributed with mean 22 oz. for a supposedly 20 oz. box. Suppose the standard deviation of the amount filled is 1.3 oz. What is the probability that a federal regulatory selects four of these boxes at random and finds that the average content of these boxes is less than 18 oz?


The Attempt at a Solution


[PLAIN]http://img508.imageshack.us/img508/3904/49450328.jpg

From the graph, I calculated the Z score using the Z score equation for distribution of sample means.

The Z-score I got was -6.15

I then found the area from Table E which came out to be 0.4999 (Table E said any value greater than 3.09 use 0.4999. Since it's to the left of the mean, I subtracted 0.5 from the Z value (0.5-0.4999) and my answer came out to be 0.001

Can anyone please double check if I got the answer correct?

I'm a bit confused by the question when it says that it's normally distributed with a mean of 22 oz for a "supposedly 20 oz box." So is it implying that the mean is 22 or 20?
They are not implying, they are saying that the mean is 22.
 
Last edited by a moderator:
  • #3
Ahhh.. The wordings of these problems are so confusing... But i guess they're meant to throw me off...

So i guess i got the answer correct then if I used 22 for the mean?
 

1. What is the Central Limit Theorem?

The Central Limit Theorem (CLT) is a fundamental concept in statistics that states that as the sample size increases, the sample mean of a random variable will approach a normal distribution, regardless of the shape of the population distribution. Essentially, it allows us to make inferences about a population based on a sample.

2. Why is the Central Limit Theorem important?

The CLT is important because it allows us to use a sample to make inferences about a population, even if the population distribution is not known. This is crucial in many applications, such as hypothesis testing and constructing confidence intervals.

3. Is the Central Limit Theorem always applicable?

No, the CLT is not always applicable. It requires certain conditions to be met, such as a large sample size (typically n ≥ 30) and the observations in the sample being independent of each other. If these conditions are not met, the CLT may not hold and other methods must be used.

4. How is the Central Limit Theorem used in real-world applications?

The CLT is used in a variety of real-world applications, such as market research, quality control, and polling. For example, a company may use the CLT to estimate the average income of their customers based on a sample, or a political poll may use the CLT to make predictions about the voting behavior of a larger population.

5. Are there any limitations to the Central Limit Theorem?

Yes, there are some limitations to the CLT. As mentioned before, it requires certain conditions to be met in order to hold. Additionally, the CLT only applies to the sample mean and may not be applicable to other statistics, such as the sample median. It also assumes that the population has a finite variance, which may not always be the case. Therefore, it is important to consider the limitations of the CLT when applying it in real-world situations.

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