Answer the following questions: A. In general, what data make up a distribution

Question

Answer the following questions: A. In general, what data make up a distribution of raw scores? What does each data point represent? B. In general, what data make up a sampling distribution of the mean? What does each data point represent? C. Suppose you have three standard deviations, all derived from the same population: the standard deviation of a population of raw scores, the standard deviation of a sample of scores, and the standard deviation of the sampling distribution of the mean (also known as the standard error). If the sample is representative of the population, and N > 1, which of the three should be the smallest in value? Why? D. Why do we need to be concerned about the shape of the sampling distribution of the mean when evaluating a null hypothesis using the z test? How is the central limit theorem useful for alleviating this concern?

Explanation / Answer

Part A

The numerical data (with or without repeated frequencies) make up a distribution of raw scores. Each data point represents the particular observation included in the distribution.

Part B

The data of the averages of the different samples of size n from the population make up a sampling distribution of the mean. Each data point represents the sample mean of the particular sample with particular sample size.

Part C

As compared to the given three different values of the standard deviations, the standard deviation for the sampling distribution of the mean should be smallest in value because for the calculation of the standard deviation of the sampling distribution of means or standard error, we divide the population standard deviation by sqrt of sample size N where N > 1, so estimate for standard deviation of the sampling distribution of means or standard error should be less than population standard deviation. It would be small than sample standard deviation also because there would be more variation in the single sample.

Part D

We know that while using Z test we assume the normal or approximate normal distribution. The sampling distribution of the means of the samples of particular sample size follows an approximate normal distribution. According to the central limit theorem, as we increase the sample size, the sampling distribution of the sample means tends to approximate normal distribution. More the sample size, better are the estimates. So, the shape of sampling distribution needs to be bell shaped or approximate normal curve for satisfying the assumption of normality for z-test.

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