Statistics Proficiency Exam Chapter 4

In a population with a mean of μ=50 and a standard deviation of σ=10, would a score of X=58 be considered an extreme value (far out in the tail of the distribution)? What if the standard deviation were σ=3?

With σ=10, a score of X=58 would be located in the central section of the distribution (within one standard deviation). With σ=3, a score of X=58 would be an extreme value, located more than two standard deviations above the mean.

_____ is distance from the mean

Deviation

The following n=12 scores are listed in order from smallest to largest. For this sample, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 7 a. Find the range and the interquartile range. b. Now, change the score of X=7 to X=27 and find the new range and interquartile range. c. Describe how one extreme score influences the range and the interquartile range.

a. The range is 7 points and the interquartile range is 2 points (from 2.5 to 4.5). b. The range is 27 points and the interquartile range is 2 points (from 2.5 to 4.5). c. The range is completely determined by the extreme scores. The interquartile range is not influenced by the most extreme scores.

A sample statistic is ________ if the average value of the statistic, for any specific sample size (n), either underestimates or overestimates the corresponding population parameter.

biased

For a sample of n scores, the _______ or _____ for the sample variance are defined as df = n - 1. The degrees of freedom determine the number of scores in the sample that are independent and free to vary.

degrees of freedom, df

What does it mean for a sample to have a standard deviation of zero? Describe the scores in such a sample.

A standard deviation of zero indicates there is no variability. In this case, all of the scores in the sample have exactly the same value.

_______ score= X-μ

Deviation

_______ = Q3- Q1

Interquartile range

The ______ is half of the interquartile range

Semi-interquartile range

What is the standard deviation for the following set of N=5 scores: 10, 10, 10, 10, and 10. (Note: You should be able to answer this question directly from the definition of standard deviation, without doing any calculations.)

Because thee is no variability (the scores are all the same), the standard deviation is zero.

______ equals the mean squared deviation

Population variance

_______ provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together

Variability

_______ is the average squared distance from the mean

Variance

Explain why the formula for sample variance divides SS by n-1 instead of dividing by n.

Without some correction, sample variability consistently underestimates the population variability. Dividing by a smaller number (n-1 instead of n) increases the value of the sample variance and makes it an unbiased estimate of the population variance.

What is the difference between a biased and an unbiased statistic?

If a statistic is biased, it means that the average value of the statistic does not accurately represent the corresponding population parameter. Instead, the average value of the statistic either overestimates or underestimates the parameter. If a statistic is unbiased, it means that the average value of the statistic is an accurate representation of the corresponding population parameter.

A set of six scores has SS=30. a. If the six scores are a population (N=6), what is the variance? b. If the six scores are a sample (n=6), what is the variance?

Population variance= 30/6= 5 Sample variance= 30/5 = 6

Consider the distribution (3, 4, 5, 7, 9, 10, 11, 13), except replace the score of 13 with a score of 100. What are the new values for the range and the semi-interquartile range? What can you conclude about these measures of variability?

Range = 98; semi-interquartile range = 3. The range is greatly affected by extreme scores in the distribution

For the following data, find the range and the semi-interquartile range: 3, 4, 5, 7, 9, 10, 11, 13

Range= URLXmax - LRLXmin = 13.5 - 2.5 = 11; semi-interquartile range = (Q3 - Q1)/2 = (10.5 - 4.5)/2 = 3

____, or _____, is the sum of the squared deviation scores

SS, sum of squares

Calculate SS, variance, and standard deviation for the following sample of n=8 scores: 0, 1, 0, 3, 6, 0, 2, 0. (Note: The definitional formula for SS works well with these scores.)

SS=32, the population variance is 4, and the standard deviation is 2.

Calculate SS, variance, and standard deviation for the following sample of n=6 scores: 11, 0, 8, 2, 4, 5. (Note: The definitional formula for SS works well with these scores.)

SS=80, the sample variance is 16 and the standard deviation is 4.

Calculate SS, variance, and standard deviation for the following sample of n=4 scores: 4, 0, 1, 1. (Note: The definitional formula for SS works well with these scores.)

SS=9, the sample variance is 3, and the standard deviation is 1.73

_______ = (Q3 - Q1)/2

Semi-interquartile range

________ = √variance

Standard deviation

Briefly explain what is measured by the standard deviation and what is measured by the variance.

Standard deviation measures the standard distance form the mean and variance measures the average square distance from the mean

A population has μ=100 and σ=20. If you select a single score from this population, o the average, how close would it be to the population mean? Explain your answer.

The standard deviation, σ=20 points, measures the standard distance between a score and the mean.

For the following scores: 3, 2, 5, 0, 1, 2, 8 a. Calculate the mean. (Note that the value of the mean does not depend on whether the set of scores is considered to be a sample or a population.) b. Find the deviation for each score, and check that the deviations add up to zero. c. Square each deviation, and compute SS. (Again, note that the value of SS is independent of whether the set of scores is a sample or a population.)

a. Mea=21/7=3.0 b and c. X, Deviation, Squared Deviation 3, 0, 0 2, -1, 1 5, +2, 4 0, -3, 9 1, -2, 4 2, -1, 1 8, +5, 25

For the data in the following sample: 1, 4, 3, 6, 2, 7, 8, 3, 7, 2, 4, 3 a. Find the median and the semi-interquartile range. b. Now change the score of X=8 to X=18, and find the new median and semi-interquartile range. c. Describe how one extreme score influences the median and semi-interquartile range.

a. Median=3.5. Semi interquartile range=2(Q1=2.5 and Q3=6.5) b. The median is still 3.5 and the semi-interquartile range is still 2 points. c. One extreme score does not influence the median or the semi-interquartile range.

There are two different formulas or methods that can be used to calculate SS. a. Under what circumstances is the definitional formula easy to use? b. Under what circumstances is the computational formula preferred?

a. The definitional formula is easy to use when the mean is a whole number and there are relatively few scores. b. The computational formula is preferred when the mean is not a whole number.

A population has a mean of μ=30 and a standard deviation of σ=5. a. If 5 points were added to every score in the population, what would be the new values for the mean and standard deviation? b. If every score in the population were multiplied by 3, what would be the new values for the mean and standard deviation?

a. The new mean is μ=35 and the standard deviation is still σ=5. b. Th new mean is μ=90 and the new standard deviation is σ=15.

A population has a mean of μ=70 and a standard deviation of σ=5. a. If 10 points were added to every core in the population, what would be the new values for the population mean and standard deviation? b. If every score in the population were multiplied by 2, what would be the new values for the population mean and standard deviation?

a. The new mean would be μ=80 but he standard deviation would still be σ=5. b. The new mean would be μ=140 and the new standard deviation would be σ=10.

For the following population of N=8 scores: 3, 3, 5, 1, 4, 3, 2, 3 a. Calculate the range, the interquartile range, and the standard deviation. b. Add 2 points to every score, then compute the range, the interquartile range, and the standard deviation again. How is variability affected by adding a constant to every score?

a. The range is 5 points, the interquartile range is 1 point (from 2.5 to 3.5), and the standard deviation is √1.25=1.12. b. After adding 2 points to every score, the range is still 4, the interquartile range is still 1, and the standard deviation is still 1.12. Adding a constant to every score does not affect variability.

A sample of n=20 scores has a mean of M=30. a. If the sample standard deviation is s=10, would a score of X=38 be considered an extreme value (out in the tail of the distribution)? b. If the sample standard deviation is s=2, would a score of X=38 be considered an extreme value (out in the tail of the distribution)?

a. With a standard deviation of 10 points, a score of X=38 would not be considered extreme. It is within one standard deviation of the mean. b. With a standard deviation of only 2 points, a score of X=38 is extreme. In this case, the score is located above the mean by a distance equal to four times the standard deviation.

The _______ is the range covered by the middle 50% of the distribution

interquartile range

The ______ is defined as the difference between the upper real limit of the largest X value and the lower real limit of the smallest X value.

range

The ______ is the distance from eh largest score to the smallest score in a distribution.

range

If the average value of the statistic is equal to the population parameter, the statistic is said to be ______. (The average value of the statistic is obtained from all the possible samples for a specific sample size, n.)

unbiased