According to internal testing done by the Get-A-Grip tire company, the lifetime of tires sold on new cars is normally distributed with a mean of 23,000 miles, with a standard deviation of 2500 miles. If the claim by Get-A-Grip is true, what is the mean of the sampling distribution of x-bar for samples of size n = 4?
X-bar = 23000
To find the mean of the sampling distribution of x-bar for samples of size n = 4, we can use the concept of central limit theorem. The central limit theorem states that for a large enough sample size, regardless of the shape of the population distribution, the sampling distribution of the sample mean will approach a normal distribution.
In this case, the population distribution follows a normal distribution with a mean of 23,000 miles and a standard deviation of 2500 miles. The sampling distribution of the sample mean will also follow a normal distribution, but with a different mean and standard deviation.
The mean of the sampling distribution of the sample mean can be found by taking the mean of the population distribution, which is 23,000 miles, and dividing it by the square root of the sample size. In this case, the sample size is 4.
So, the mean (μx-bar) of the sampling distribution of the sample mean for samples of size n = 4 is:
μx-bar = μ / √n
μx-bar = 23,000 / √4
μx-bar = 23,000 / 2
μx-bar = 11,500
Therefore, the mean of the sampling distribution of x-bar for samples of size n = 4 is 11,500 miles.