Math statistics

Question

One hundred teachers attended a seminar on mathematical problem solving. The attitudes of representative sample of 12 of the teachers were measured before and after the seminar. A positive number for change in attitude indicates that a teacher's attitude toward math became more positive. The twelve change scores are as follows.

3; 8; −1; 2; 0; 4; −3; 1; −1; 5; 4; −2

Part (a)

1. What is the mean change score? (Round your answer to two decimal places.)
2.  

Part (b)

1. What is the standard deviation for this sample? (Round your answer to two decimal places.)
2.  

Part (c)

1. What is the median change score? (Round your answer to one decimal place.)
2.  

Part (d)

1. Find the change score that is 2.2 standard deviations below the mean. (Round your answer to one decimal place.)
2.  

Answer 1

To solve this problem, we'll perform the following calculations:

(a) Mean change score:

To find the mean change score, we sum up all the change scores and divide by the total number of scores.

Change scores: 3, 8, -1, 2, 0, 4, -3, 1, -1, 5, 4, -2

Mean = (3 + 8 - 1 + 2 + 0 + 4 - 3 + 1 - 1 + 5 + 4 - 2) / 12

Mean = 32 / 12

Mean ≈ 2.67

Therefore, the mean change score is approximately 2.67.

(b) Standard deviation:

To find the standard deviation, we need to calculate the variance first. The variance is the average of the squared differences from the mean.

Change scores: 3, 8, -1, 2, 0, 4, -3, 1, -1, 5, 4, -2

Variance = [(3 - 2.67)^2 + (8 - 2.67)^2 + (-1 - 2.67)^2 + (2 - 2.67)^2 + (0 - 2.67)^2 + (4 - 2.67)^2 + (-3 - 2.67)^2 + (1 - 2.67)^2 + (-1 - 2.67)^2 + (5 - 2.67)^2 + (4 - 2.67)^2 + (-2 - 2.67)^2] / 12

Variance = (0.4489 + 28.2249 + 17.9521 + 0.1089 + 7.1289 + 1.7289 + 26.1025 + 1.7289 + 17.9521 + 5.2889 + 1.7289 + 21.5729) / 12

Variance ≈ 130.52 / 12

Variance ≈ 10.8767

Standard deviation = √Variance

Standard deviation = √10.8767

Standard deviation ≈ 3.296

Therefore, the standard deviation for this sample is approximately 3.296.

(c) Median change score:

To find the median change score, we need to arrange the change scores in ascending order and find the middle value.

Change scores: -3, -2, -1, -1, 0, 1, 2, 3, 4, 4, 5, 8

Since there are 12 scores, the middle value is the average of the 6th and 7th scores.

Median = (1 + 2) / 2

Median = 3 / 2

Median = 1.5

Therefore, the median change score is 1.5.

(d) Change score 2.2 standard deviations below the mean:

To find the change score that is 2.2 standard deviations below the mean, we subtract 2.2 times the standard deviation from the mean.

Change score = Mean - (2.2 * Standard deviation)

Change score = 2.67 - (2.2 * 3.296)

Change score = 2.67 - 7.2512

Change score ≈ -4.5812

Therefore, the change score that is 2.2 standard deviations below the mean is approximately -4.6.