A random variable, X, has a mean of 10 and a standard deviation of 15. Another random variable, Y, is defined by Y = 2X + 3. Then, which of the following is/are true? I. The mean of Y is 33. II. The variance of Y is 30. III. The standard deviation of Y is 33. IV. X and Y are perfectly correlated.
Answer(s): E
The mean of Y is 2*10 + 3 = 23. The standard deviation of Y is 2*15 = 30. Its variance is then equal to 30^2 = 900. Finally, since Y is a linear multiple of X, X and Y are perfectly correlated (positively, in this case)
A systematic sampling method should not be used when:
Answer(s): C
If the data are already arranged in the form of a predetermined pattern, using systematic sampling can introduce a systematic pattern in the sample itself, which would lead to erroneous inferences about the entire population.
What is the relationship between the variance and the standard deviation?
The variance is the mean of the squared deviations from the mean. The standard deviation is the positive square root of the variance.
A bell-shaped, symmetrical frequency distribution has a mean of 5 and a standard deviation of 2.5. The percentage of observations less than zero is about:
Answer(s): D
Note that the mean is 2 standard deviations above zero. The fraction of observations in a bell-shaped, symmetrical frequency distribution which lie outside the 2-sigma range around the mean is about 5%. Since the distribution is symmetrical about the mean, the percentage of observations less than zero is approximately equal to 5%/2 = 2.5%.
What is the area under the normal curve between z = 1.0 and z = 2.0?
Answer(s): B
From the z-tables, z = 1 is 0.3413 and z = 2 is 0.4772. So the area in between is 0.4772 - 0.3413 = 0.1359.
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