For an IQ test, we know the population _ 5 100 and the _X 5 16. We are interested in creating the sampling distribution when N 5 64. (a) What does that sampling distribution of means show? (b) What is the shape of the distribution of IQ means and the mean of the distribution? (c) Calculate _X for this distribution. (d) What is your answer in part (c) called, and what does it indicate? (e) What is the relative frequency of sample means above 101.5?
Population mean = 100 and the standard deviation (SD) = 16, and N = 64?
Do you know where the = sign is?
Even so, we do not have data on the sample.
(a) The sampling distribution of means shows the distribution of all possible sample means that can be obtained from a population. In this case, we are interested in creating the sampling distribution of means when the sample size (N) is 64.
(b) In general, when the sample size is large (N ≥ 30) and the population is normally distributed or the sample size is even larger (N ≥ 100), the sampling distribution of means will be approximately normally distributed. The shape of the distribution of IQ means will likely be close to a normal distribution with a mean approximately equal to the population mean (_X = 16).
(c) To calculate _X for the distribution, we use the formula: _X = _X / N.
(_X is the population mean, and N is the sample size.)
In this case, _X = 16 (population mean) and N = 64 (sample size). Plugging these values into the formula, we get:
_X = 16 / 64 = 0.25
Therefore, _X for this distribution is 0.25.
(d) The answer from part (c) is called the "sample mean" or "estimated population mean." It indicates the average value or central tendency of the sampling distribution of means.
(e) To determine the relative frequency of sample means above 101.5, we need more information about the population standard deviation. With just the population mean and sample size, it's not possible to calculate the exact relative frequency.