Suppose it is known that the average score of all high school seniors taking a national test of basic skills is 420. These scores are normally distributed with a standard deviation of 80. A school gives the test to a sample of 144 seniors.
Identify the following characteristics of the sampling distribution for the sample mean (x-bar).
i. the mean
ii. the standard deviation
iii. the shape
To find the characteristics of the sampling distribution for the sample mean, we need to use the properties of the normal distribution.
i. The mean (μ) of the sampling distribution for the sample mean (x-bar) will be the same as the mean of the population, which is given as 420.
ii. The standard deviation (σ) of the sampling distribution for the sample mean (x-bar) can be calculated using the formula:
σ(x-bar) = σ / √n
where σ is the standard deviation of the population and n is the sample size. In this case, the standard deviation of the population (σ) is given as 80, and the sample size (n) is 144. Plugging in these values, we get:
σ(x-bar) = 80 / √144
= 80 / 12
= 6.67.
So, the standard deviation of the sampling distribution for the sample mean is 6.67.
iii. The shape of the sampling distribution for the sample mean (x-bar) is known to be approximately normal when the sample size is sufficiently large (typically n ≥ 30), regardless of the shape of the population distribution. In this case, the sample size is 144, which is larger than 30, so we can assume that the shape of the sampling distribution for the sample mean will be approximately normal.