Chapter 5 & 6 essay
On a psychology exam with µ = 76 and σ = 12, Tom scored 9 points below the mean, Mary had a score of X = 70, and Bill had a z score of z = -1.00. Place these three students in order from lowest to highest score. _______________ Student with lowest score _______________ Student with middle score _______________ Student with highest score
Bill: X = 64, z = -1.00 Tom: X = 67, z = -0.75 Mary: X = 70, z = -0.50
On a psychology exam with µ = 76 and σ = 12, Tom scored 9 points below the mean, Mary had a score of X = 70, and Bill had a z score of z = -1.00. Place these three students in order from lowest to highest score. _______________ Student with lowest score _______________ Student with middle score _______________ Student with highest score
Bill: X = 64, z = -1.00 Tom: X = 67, z = -0.75 Mary: X = 70, z = -0.50
For a distribution of scores, X = 40 corresponds to a z score of z = +1.00, and X = 28 corresponds to a z score of z = -0.50. What are the values for the mean and standard deviation for the distribution? (Hint: Sketch a distribution and locate each of the z score positions.)
The 12 points between the two scores corresponds to a total of 1.5 standard deviations. Therefore, σ = 8 and µ = 32.
Describe the general purpose of a z score and explain how a z-score accomplishes this goal.
The purpose of a z score is to describe a location within a distribution using a single number. The z-score converts each X value into a signed number so that the sign tells whether the score is located above (+) or below (-) the mean, and the number identifies the distance from the mean by measuring the number of standard deviations between the score and the mean.
Describe what happens to the mean, the standard deviation, and the shape of a distribution when all of the scores are transformed into z scores.
When an entire distribution of scores is transformed into z scores, the resulting distribution will have a mean of zero, a standard deviation of one, and the same shape as the original distribution.
For a population with µ = 48 and σ = 8, find the X value that corresponds to each of the following z scores: -0.25, -1.50, 0.50, 2.00
X = 46 (2 points below the mean) X = 36 (12 points below the mean) X = 52 (4 points above the mean) X = 64 (16 points above the mean
A population of scores with µ = 73 and σ = 20 is standardized to create a new population with µ = 50 and σ = 10. What is the new value for each of the following scores from the original population? Scores: 63, 65, 77, 83
X = 63 -> z = -0.50 -> X = 45 X = 65 -> z = -0.40 -> X = 46 X = 77 -> z = +0.20 -> X = 52 X = 83 -> z = +0.50 -> X = 55
Assume that a vertical line is drawn through a normal distribution at each of the following z-score locations. In each case, determine whether the tail is on the left side or the right side of the line and find the proportion of the distribution that is located in the tail.
a. The tail is on the right. p = 0.0359 b. The tail is on the right. p = 0.2743 c. The tail is on the left. p = 0.3446 d. The tail is on the left. p = 0.1056
A normal distribution has a mean of µ = 100 with σ = 20. Find the following probabilities: a. p(X > 102) c. p(X < 130) b. p(X < 65) d. p(95 < X<105)
a. z = 0.10, p = 0.4602 b. z = -1.75, p = 0.0401 c. z=1.50, p = 0.9332 d. -0.25 < z < +0.25, p = 0.1974
Assume that the total score (from both teams) for college football games averages µ = 42 points per game, and that the distribution of total points is approximately normal with σ = 20. a. What is the probability that a randomly selected game would have more than 60 points? b. What proportion of college football games have a point total between 20 and 60?
a. z = 0.90, p = 0.1841 b. -1.10 < z < 0.90, p = 0.6802
In an ESP experiment subjects must predict whether a number randomly generated by a computer will be odd or even. a. What is the probability that a subject would guess exactly 18 correct in a series of 36 trials? b. What is the probability that a subject would guess more than 20 correct in a series of 36 trials?
a. With n = 36 and p = q = 1/2, you may use the normal approximation with µ = 18 and o = 3. X = 18 has real limits of 17.5 and 18.5 corresponding to z = -0.17 and z = +0.17. p = 0.1350. b. p(X > 20.5) = p(z > 0.83) = 0.2033
For a normal distribution, a. What z-score separates the highest 10% from the rest of the scores? b. What z-score separates the highest 30% from the rest of the scores? c. What z-score separates the lowest 40% from the rest of the scores? d. What z-score separates the lowest 20% from the rest of the scores?
a. z = 1.28 b. z = 0.52 c. z = -0.25 d. z = -0.84
For a normal distribution with µ = 200 and σ = 50, find the following values: a. What X value separates the highest 10% of the distribution from the rest of the scores? b. What X values form the boundaries for the middle 60% of the distribution? c. What is the probability of randomly selecting a score greater than X = 325?
a. z = 1.28, X = 264 b. z = 0.84, X = 158 to 242 c. z = 2.50, p = 0.0062
For a population with µ = 60 and σ = 12, find the z score corresponding to each of the following X values: 66, 78, 57, 48
z = +0.50 (above the mean by ½ standard deviation) z = +1.50 (above the mean by 1 ½ standard deviations) z = -0.25 (below the mean by ¼ standard deviation) z = -1.00 (below the mean by 1 standard deviation)
