For a normal distribution with a mean of 140 and a standard deviation of 55 find each value requested.

a. What is the minimum score needed to be in the bottom 10% of the distribution?

b. What scores form the boundaries for the middle 75% of the distribution.

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion (.10) related to the Z score. Insert values to find score.

Middle 75% = ± .375 between score and mean. Use the same table.

To find the answer to these questions, you can use the standard normal distribution table or a statistical calculator.

a. To find the minimum score needed to be in the bottom 10% of the distribution, you need to find the z-score corresponding to the 10th percentile.

1. Convert the raw score to a z-score using the formula:
z = (x - μ) / σ

Where:
x = raw score
μ = mean
σ = standard deviation

2. Look up the z-score in the standard normal distribution table or use a statistical calculator to find the corresponding percentile.

b. To find the scores that form the boundaries for the middle 75% of the distribution, you need to find the z-scores corresponding to the 12.5th and 87.5th percentiles.

1. Convert the raw scores to z-scores using the formula mentioned above.

2. Use the z-score values to find the corresponding raw scores by rearranging the formula:
x = μ + (z * σ)

Where:
x = raw score
μ = mean
σ = standard deviation
z = z-score

3. Use the obtained raw scores as the boundaries for the middle 75% of the distribution.

If you have access to a statistical calculator or a standard normal distribution table, you can directly find the required percentiles and use the formulas to convert z-scores to raw scores.