The number of column inches of classified advertisements appearing on Mondays in a certain daily
newspaper has mean 320 inches and standard deviation 30 inches. Suppose that the results for 100
consecutive Mondays can be regarded as a random sample and let J denote the mean number of column
inches of classified advertisements in the sample. Assuming a sample of 100 is sufficiently large, the
random variable J has a
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The random variable J, representing the mean number of column inches of classified advertisements in the sample, follows an approximately normal distribution. This is known as the sampling distribution of the sample mean.
To understand how to calculate the parameters of this sampling distribution, we can look at the properties of the population distribution and the Central Limit Theorem.
1. Population distribution:
Based on the information given, the population distribution of the number of column inches of classified advertisements on Mondays in the daily newspaper has a mean of 320 inches and a standard deviation of 30 inches. We assume this distribution is approximately normal.
2. Central Limit Theorem:
The Central Limit Theorem states that, for a sufficiently large sample size, the distribution of the sample means (J) will be approximately normal, regardless of the shape of the population distribution. In other words, the sampling distribution of the mean becomes more bell-shaped as the sample size increases.
To calculate the parameters of the sampling distribution, we need to determine the mean and standard deviation of the sample mean:
1. Mean of the sampling distribution (µJ):
The mean of the sampling distribution of sample means is equal to the mean of the population, which is 320 inches.
2. Standard deviation of the sampling distribution (σJ):
The standard deviation of the sampling distribution of sample means, also known as the standard error, can be calculated using the formula:
σJ = σ / √n
where σ is the standard deviation of the population (30 inches) and n is the sample size (100).
Substituting the given values into the formula, we get:
σJ = 30 / √100
σJ = 30 / 10
σJ = 3 inches
Therefore, for a sample size of 100, the random variable J, representing the mean number of column inches of classified advertisements in the sample, has a sampling distribution with a mean of 320 inches and a standard deviation (standard error) of 3 inches.