Suppose that the packaging equipment in a manufacturing process that is filling 368 gram packages
of cereal is set so that the amount of cereal in the box is normally distributed with mean of 368
grams. Frompast experience, the standard deviation of the population is known to be 15 grams.
(a) What can you say about the sampling distribution of the sample mean of the box weight if a
sample of 25 boxesis taken from boxesfilled by this equipment?
can anyone help me understand this question?
Of course! This question is asking about the sampling distribution of the sample mean of the box weight from a manufacturing process that fills packages of cereal.
To understand this question better, let's break down some key concepts:
1. Sampling Distribution: In statistics, a sampling distribution refers to the probability distribution of a sample statistic. In this case, we are interested in the sample mean of the box weight.
2. Sample Mean: The sample mean is the average value of a sample. It is calculated by summing up all the values in a sample and then dividing by the sample size.
Now, given this background, let's focus on the specific details provided in the question:
- The packaging equipment is set to fill 368-gram packages of cereal.
- The population (all the boxes filled by this equipment) follows a normal distribution with a mean of 368 grams.
- The standard deviation of the population is known to be 15 grams.
- A sample of 25 boxes is taken.
(a) Now, we need to determine what we can say about the sampling distribution of the sample mean based on this information.
The central limit theorem states that when a sample size is sufficiently large (usually 30 or more), the sampling distribution of the sample mean will approximate a normal distribution, regardless of the shape of the population distribution.
In this case, the sample size is 25, which is less than 30. However, if the population distribution is already normally distributed (as stated in the question), we can assume that the distribution of the sample mean will also be normal, regardless of the sample size.
Therefore, we can say that the sampling distribution of the sample mean of the box weight will be approximately normal, with a mean of 368 grams (same as the population mean) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (in this case, 25).
To calculate the standard deviation of the sampling distribution, we divide the population standard deviation (15 grams) by the square root of the sample size:
Standard Deviation of Sampling Distribution = Population Standard Deviation / √(Sample Size)
= 15 grams / √(25)
= 15 grams / 5
= 3 grams
So, the sampling distribution of the sample mean will be approximately normal, with a mean of 368 grams and a standard deviation of 3 grams.