Question

the amounts of electricity bills for all households in a particular city have an approximately normal distribution with a mean of 140.00 and a standard deviation of 30.00. let line over x be the mean amount of electricity bills for a random sample of 25 households selected from this city. Find the mean and standard deviation of line over x and comment on the shape of its sampling distribution

Answer 1

The mean of the sampling distribution of the sample mean (line over x) is $140.00, the same as the population mean. The standard deviation (standard error) of line over x is $6.00. Due to the Central Limit Theorem, the shape of this sampling distribution is approximately normal.

Step-by-step explanation:

The question is about the sampling distribution of the sample mean, denoted as a line over x, for a normally distributed population of electricity bills. The mean (μ) of the population is given as $140.00, and the standard deviation (σ) is given as $30.00. When taking a random sample of size 25 from this population, the mean of the sampling distribution of line over x remains the same as the population mean, which is $140.00.

The standard deviation of the sampling distribution of line over x often called the standard error (SE), is calculated by dividing the population standard deviation by the square root of the sample size. In this case, SE = σ / √n = $30.00 / √25 = $30.00 / 5 = $6.00. The shape of the sampling distribution will also be approximately normal because of the Central Limit Theorem, which states that the distribution of sample means will be approximately normal if the sample size is large enough (n≥25 in most cases).