Statistics Proficiency Exam Chapter 5

For a sample with a standard de aviation of s= 6, a score fo X= 65 corresponds to z= 1.50. What is the sample mean?

M= 56

_______ are used to make dissimilar distributions comparable.

Standardized distributions

What is the advantage of having a mean of μ=0 for a distribution of z-scores?

With a mean of zero, all positive scores are above the mean and all negative scores are below the mean

For a distribution with μ=40 and σ=8, find the X value corresponding to each of the following z-scores. a. z=1.50 b. z=-1.25 c. z=0.50

a. X=52 b. X=30 c. X=44

A population consists of the following N= 6 scores: 6, 1, 3, 4, 7, and 3. a. Compute μ and σ for the population. b. Find the z-score for each score in the population. c. Transform the original population into a new population of N= 6 scores with a mean of μ= 50 and a standard deviation of σ= 10.

a. μ= 3 and σ= 2 b and c. Original X, z-score, Transformed X 6, 1.00, 60 1, -1.50, 35 3, -0.50, 45 4, 0, 50 7, 1.50, 65 3, -0.50, 45

For a sample with a mean of M=36, a score of X=40 corresponds to z=0.50. What is the standard deviation for the sample?

s=8

In a distribution of scores, X= 62 corresponds to z= +0.50, and X= 52 corresponds to z= -2.00. Find the mean and standard deviation for the distribution.

μ= 60 and σ= 4. The distance between the two scores is 10 points, which is equal to 2.5 standard deviations.

In a distribution with σ=12, a score of X=56 corresponds to z=-0.50. What is the mean for this distribution?

μ=62

A distribution of English exam scores has μ=70 and σ=4. A distribution of history exams has μ=65 and σ=15. For which exam would a score of X=78 have a higher standing? Explain your answer.

For the English exam, X=78 corresponds to z=2.00, which is a higher standing than z=13/15=0.87 for the history exam.

For a sample with a standard deviation of s=12, a score of X=83 corresponds to z= -0.25. What is the mean for the sample?

M=86

A sample has a mean of M= 80 and a standard deviation of s= 5. For this sample, find the X value corresponding to each of the following z-scores. z= 0.80 z= 1.20 z= 2.00 z= -0.40 z= -0.60 z= -1.80

X, z 84, 0.80 86, 1.20 90, 2.00 78, -0.40 77, -0.60 71, -1.80

A score that is 4 points above the mean corresponds to a z-score of z= 0.50. What int eh population standard deviation?

σ= 8

In a distribution with μ=50, a score of X=48 corresponds to z=-0.50. What is the standard deviation for this distribution?

σ=4

The numerical value of the _______ specifies the distance from eh mean by counting the number of standard deviations between X and μ.

z-score

The sign of the ______ (+ or -) signifies whether the score is above the mean (positive) or below the mean (negative).

z-score

For a sample with a mean of M=20 and a standard deviation of s=4, find the z-score correspond to each of the following X values. X=12 X=14 X=19 X=20 X=22 X=30

z= -2.00 z= -1.50 z= -0.25 z= 0 z= 0.50 z= 2.50

Suppose that you have a score of X= 75 on an exam with μ= 70. Which standard deviation would give you a better grade: σ= 2 or σ= 10?

σ= 2

For a population with a mean of μ= 70, a score of X= 64 corresponds to z= -2.00. What is the population standard deviation?

σ= 3

What information is provided by the sign (+/-) of a z-score? What information is provided by the numerical value of the z-score?

A z-score describes a precise location within a distribution. The sign of the z-score tells whether the location is above (+) or below (-) the mean, and the magnitude tells the distance from the mean in terms of the number of standard deviations.

A normal-shaped distribution with μ=40 and σ=8 is transformed into z-scores. The resulting distribution of z-scores will have a mean of ______ and a standard deviation of _____.

A z-score distribution always has a mean of 0 and a standard deviation of 1.

For a population with μ= 60 and σ= 9, find the z-score for each of the following X values. (Note: You probably will need to use a formula and a calculator to find these values.) X= 65 X= 70 X= 53 X= 48 X= 75 X= 58

X, z 65, 0.56 70, 1.11 53, -0.78 48, -1.33 75, 1.67 58, -0.22

Which of the following exam scores should lead to the better grade? a. A score of X=43 on an exam with μ= 40 and σ= 2. b. A score of X= 60 on an exam with μ= 50 and σ= 20

X= 43 corresponds to z= 1.50, and X= 60 corresponds to z= 0.50. X= 43 has a higher position in its distribution and should receive the higher grade.

Find the X value corresponding to z= 1.50 for each of the following distributions. a. μ= 40 and σ= 6 b. μ= 40 and σ= 20 c. μ= 80 and σ= 4 d. μ= 80 and σ= 8

X= 49 X= 70 X= 86 X= 92

For a sample with a mean of M=80 and a standard deviation of s=10, find the X value corresponding to each of the following z-scores. z= -1.00 z= -0.50 z= -0.20 z= 1.50 z= 0.80 z= 1.40

X= 70 X= 75 X= 78 X= 95 X= 88 X= 94

The following sample of n= 6 scores has a mean of M= 6 and a standard deviation of s= 4. Scores: 10, 0, 4, 8, 4, and 10. a. Find the z-score for each X value. b. For the sample of n= 6 z-scores, verify that the mean is zero and the standard deviation is 1.

a. X, z 10, +1.00 0, -1.50 4, -0.50 8, +0.50 4, -0.50 10, +1.00 b. For the sample of n= 6 z-scores, Σz= 0 and M= 0. Also for the z-scores, SS= 5, s²= 5/5= 1, and s= 1.

A population of N=6 scores has μ=8 and σ=5. The six scores are 14, 8, 0, 11, 3, and 12. a. Using z-scores, transform the population into a new distribution with μ=100 and σ=20. (Find the new score corresponding to each of the original scores.) b. Compute the mean and standard deviation for the new scores. (You should obtain μ=100 and σ=20.)

a. The original scores correspond to z-scores of 1.20, 0, -1.60, 0.60, -1.00, and 0.80. These values are transformed into new scores of 124, 100, 68, 112, 80, and 116. b. The new scores add to ΣX=600 so the new mean is μ=100. The SS for the transformed scores is SS=2400, the variance is 400, and the new standard deviation is σ=20.

A population has a mean of μ= 40 and a standard deviation of σ= 12. a. For the population, find the z-score for each of the following X values. X= 43 X= 49 X= 52 X= 34 X= 28 X= 64 b. For the same population, find the score (X value) that corresponds to each of the following z-scores. z= 0.75 z= 1.50 z= -2.00 z= -0.25 z= -0.50 z= 1.25

a. X, z 43, 0.25 49, 0.75 52, 1.00 34, -0.50 28, -1.00 64, 2.00 b. X, z 49, 0.75 58, 1.50 16, -2.00 37, -0.25 34, -0.50 55, 1.25

A sample has a mean of M=30 and a standard deviation of s=8. a. Would a score of X=36 be considered a central score or an extreme score in the sample? b. If the standard deviation were s=2, would X=36 be central or extreme?

a. X= 36 is a central score corresponding to z= 0.75 b. X= 36 is an extreme score corresponding to z= 3.00

A distribution with a mean of μ= 64 and a standard deviation of σ= 4 is being transformed into a standardized distribution with μ= 100 and σ= 20. Find the new, standardized score for each of the following values form the original population. a. X= 60 b. X= 52 c. X= 72 d. X= 66

a. X= 80 (z= -1.00) b. X= 40 (z= -3.00) c. X= 140 (z= 2.00) d. X= 110 (z= 0.50)

For a population with μ=60 and σ=20, find the X value corresponding to each of the following z-scores: a. z=-0.25 b. z=2.00 c. z=0.50

a. X=55 b. X=100 c. X=70

A distribution has a standard deviation of σ= 4. Describe the location of each of the following z-scores in terms of its position relative to the mean. For example, z= +1.00 is a location that is 4 points above the mean. a. z= +2.00 b. z= +0.50 c. z= -2.00 d. z= -0.50

a. above the mean by 8 points b. above the mean by 2 points c. below the mean by 8 points d. below the mean by 2 points

For a distribution with μ=40 and σ=8, find the z-score for each of the following scores. a. X=36 b. X=46 c. X=56

a. z=-0.50 b. z=0.75 c. z=2.00

For a population with μ=20 and σ=4, find the z-score for each of the following scores: a. X=18 b. X=28 c. X=20

a. z=-0.50 b. z=2.00 c. z=0

Identify the z-score value corresponding to each of the following location sin a distribution. a. Below the mean by 1 standard deviation. b. Above the mean by 1.5 standard deviation. c. Below the mean by 3/4 standard deviation.

a. z=-1.00 b. z=1.5 c. z=-0.75

A population of scores has μ=73 and σ=8. IF the distribution is standardized to create a new distribution with μ=100 and σ=20, what are the new values for each of the following scores from the original distribution? a. X=65 b. X=71 c. X=81 d. X=83

a. z=-1.00, X=80 b. z=-0.25, X=95 c. z=1.00, X=120 d. z=1.25, X=125

A _______ is composed of scores that have been transformed to create predetermined values for μ and σ.

standardized distribution

A ______ specifies the precise location of each X value within a distribution.

z-score