We have a group of 75 people and for their raw scores, you are given the Mean (M = 30) and variance (VAR = 16). Assuming a normal distribution of scores, what is the probability of randomly selecting a score between 27 and 31 for this group?

To calculate the probability of randomly selecting a score between 27 and 31 for this group, we need to use the concept of standard deviation.

The standard deviation (SD) can be calculated using the variance (VAR) by taking the square root of it (SD = √VAR). In this case, the standard deviation would be √16 = 4.

Since the distribution of scores is assumed to be normal, we can use the Z-score formula to standardize the range (27 to 31). The Z-score is calculated by subtracting the mean from the score and then dividing by the standard deviation.

For the lower value, the Z-score would be (27 - 30) / 4 = -3/4 = -0.75.
For the upper value, the Z-score would be (31 - 30) / 4 = 1/4 = 0.25.

Now, we need to find the probability associated with these Z-scores. This can be done by referring to the Z-table or by using a statistical software or calculator.

From the Z-table or statistical software, we find that the probability associated with a Z-score of -0.75 is approximately 0.2266, and the probability associated with a Z-score of 0.25 is approximately 0.5987.

To find the probability of the score falling between 27 and 31, we subtract the lower probability from the upper probability: 0.5987 - 0.2266 = 0.3721, or approximately 37.21%.

Therefore, the probability of randomly selecting a score between 27 and 31 for this group is approximately 37.21%.