For an IQ test, we know the population mean = 100 and the standard deviation =16. We are interested in creating the sampling distribution when N = 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 standard deviation 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?
To answer these questions, we need to understand the concept of sampling distribution. A sampling distribution is a probability distribution of a sample statistic based on repeated sampling from the same population. In this case, we want to create the sampling distribution of means based on IQ scores.
(a) The sampling distribution of means shows the distribution of means that we would expect to obtain from all possible samples of a specific size (N = 64) from the given population. Each sample will have a mean IQ score calculated, and the sampling distribution of means represents the distribution of these sample means.
(b) The shape of the distribution of IQ means can be approximated to a normal distribution, also known as a Gaussian distribution. The mean of the distribution of IQ means will be equal to the population mean, which is 100 in this case.
(c) To calculate the standard deviation for this distribution, we need to use the formula for the standard deviation of the sample means, also known as the standard error (SE) of the mean. The formula is:
SE = population standard deviation / square root of sample size
Given that the population standard deviation is 16 and the sample size is 64, we can calculate the standard error as follows:
SE = 16 / √64 = 16 / 8 = 2
Therefore, the standard deviation for this distribution is 2.
(d) The answer to part (c), which is the standard deviation for this distribution, is called the standard error of the mean. It indicates the average amount of variability or dispersion we would expect to see in the sample means when sampling from the given population.
(e) To determine the relative frequency of sample means above 101.5, we need to use the concept of z-scores and the standard normal distribution. The z-score represents the number of standard deviations a particular sample mean is above or below the mean of the sampling distribution.
To calculate the z-score for a sample mean of 101.5, we use the formula:
z = (x - mean) / standard error
Where x is the sample mean, mean is the population mean (100), and the standard error is 2.
So, the z-score is:
z = (101.5 - 100) / 2 = 1.5 / 2 = 0.75
Now, we can look up the area under the standard normal distribution curve for a z-score of 0.75. Using a standard normal table or a calculator, we find that the area to the right of a z-score of 0.75 is approximately 0.2266.
Therefore, the relative frequency of sample means above 101.5 is approximately 0.2266, or 22.66%.