A set of scores with a normal distribution has a mean of 32 and a standard deviation of 3.7. Which score could be expected to occur the least often?
To determine which score could be expected to occur the least often in a normal distribution, you can look at the z-scores associated with each individual score. The z-score measures the number of standard deviations a data point is away from the mean.
In a normal distribution, the scores that are farther away from the mean are less likely to occur. This means that the score with the highest z-score, either positive or negative, would be expected to occur the least often.
To find the z-score for a specific score, you can use the formula:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the individual score
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
In this case, the mean (μ) is 32 and the standard deviation (σ) is 3.7. Let's calculate the z-score for the highest and lowest scores to determine which would be expected to occur the least often.
Since we want to find the score that is expected to occur the least often, we need to find the score with the highest positive or negative z-score. The z-score will be higher for scores that are farther away from the mean.
Now, let's calculate the z-score for the highest and lowest possible scores.
For the highest score:
z = (x - μ) / σ
z = (x - 32) / 3.7
For the lowest score:
z = (x - μ) / σ
z = (x - 32) / 3.7
To determine the score that is expected to occur the least often, we need to compare the absolute values of the z-scores. The score with the highest absolute z-score will be the one expected to occur the least often.
Therefore, we need to calculate the z-score for the highest possible score and the z-score for the lowest possible score using the formula mentioned above.
Hence, the score with the highest absolute z-score will be the one expected to occur the least often in the set of scores with a normal distribution.