Chapter 12 short answer questions -- KEY
6. What are the characteristics of the sampling distribution of the mean?
The sampling distribution of the mean is a distribution of a population of sample means. This distribution has a mean and a standard deviation, and because the distribution of sample means is a population distribution, the mean and standard deviation are population parameters. The mean of the sampling distribution is equal to m , the mean of the population of raw scores from which the means were sampled. The standard deviation of the sampling distribution of means (known as the standard error of the mean) is equal to the standard deviation of the population of raw scores divided by the square root of the sample size, s /Ö N.
7. Explain why the standard deviation of the sampling distribution of the mean is sometimes referred to as the "standard error of the mean."
The sampling distribution of the mean is the distribution of the mean of all possible samples of size N that can be drawn from the raw score population. Because each sample’s mean is an estimate of the population mean and because sampling variability will cause the sample means to vary from the population mean, each sample mean is only an approximation of the population mean. The term standard error thus refers to the sampling error that results when sample means are used to estimate the population mean.
11. Is the shape of the sampling distribution of the mean always the same as the shape of the null hypothesis population? Explain.
No, it is not always the same. The shape of the sampling distribution of the mean becomes more normal when sample sizes are larger, no matter what the shape of the null hypothesis population. Thus, when the null hypothesis population is normally distributed, then the sampling distribution of the mean will have the same shape. However, when the null hypothesis population has a shape that is non-normal, the sampling distribution of the mean will be similar to the null hypothesis distribution only when sample sizes are very small. As the sample size gets larger, the shape of the sampling distribution of the mean gets more normal and more different from the null hypothesis population’s shape.
12. In using the z-test, why is important that the sampling distribution of the mean be normally distributed?
If the sampling distribution of the mean is NOT normally distributed, then the probabilities given for areas beyond a z score (shown in column c in the z-tables at the end of the textbook) are not valid. It is only because the shape of the normal curve is exactly known that those probabilities are accurate. If we do NOT know that the sampling distribution is normally shaped, then we don’t know, for example, that 1.96 cuts off the upper 2.5% of the scores (means) in the distribution.
13. If the assumptions underlying the z test are met, what are the characteristics of the sampling distribution of z?
The assumptions underlying the z test are that the raw score population mean and standard deviation are known and that the sampling distribution of sample means is normally shaped. This second assumption is met whenever the raw score population is normally shaped or when sample sizes are sufficiently large. If these assumptions are met, then the sampling distribution of z is normally shaped, it has a mean of 0 and a standard deviation of 1.