Reading readiness of preschoolers from an impoverished neighborhood (n=20) was measured using a standardized test. Nationally, the mean on this test for preschoolers is 30.9 with SD=2.08.

A) Children below the 30th percentile (in the bottom 30%) are in need of special assistance prior to attending school. What raw score marks the cut-off score for these children?

B) What percentage of children score between 25 and 28.5?

C) How many children would we expect to find with scores between 28 and 31.5?

D) Children in the top 25% are considered accelerated readers and qualify for different placements in school. What raw score would mark the cutoff for such placement?

#2. Age at onset of dementia was determined for a sample of adults between the ages of 60 and 75. For 15 subjects, the results were EX=1008, and E(X-M)^2 = 140.4. Use this information to answer the following:

A) What is the mean and SD for this data?

B) Based on the data you have and the Normal Curve Tables, what percentage of people might start to show signs of dementia at or before age 62?

C) If we consider the normal range of onset in this population to be +/-1 z-score from the mean, what two ages correspond to this?

D) A neuropsychologist is interested only in studying the most deviant portion of this population, that is, those individuals who fall within the top 10% and the bottom 10% of the distribution. She must determine the ages that mark these boundaries. What are these ages?

#1. Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions that correspond to the Z scores. For D, reverse process and insert Z score in equation above.

#2. A) Assuming EX = mean of sample, divide E(X-M)^2 by the number of scores to get variance.

Standard deviation = square root of variance

B, C, D) Use the table and equation above.

I'll let you do the calculations.

67.2 = mean

10.03= SD
percentage=48.81%

A) To find the raw score that marks the cutoff for children below the 30th percentile, we need to use the z-score formula.

Z-score formula: (X - μ) / σ

First, we calculate the z-score for the 30th percentile using the formula:
Z = (X - μ) / σ

Z = (X - 30.9) / 2.08

Since we want to find the raw score (X) that corresponds to the 30th percentile, we can rearrange the formula:

X = Z * σ + μ

For the 30th percentile, the z-score can be found by referencing the z-table or using a statistical calculator. The z-score that corresponds to the 30th percentile is approximately -0.52.

Plugging in the values into the rearranged formula:

X = -0.52 * 2.08 + 30.9

X ≈ 29.82

Therefore, the raw score cutoff for children below the 30th percentile is approximately 29.82.

B) To find the percentage of children who score between 25 and 28.5, we need to calculate the z-scores for these two values and then find the corresponding areas under the standard normal curve.

First, calculate the z-scores for 25 and 28.5:

Z1 = (25 - 30.9) / 2.08
Z2 = (28.5 - 30.9) / 2.08

Using the z-table or a statistical calculator, we can find the area under the standard normal curve corresponding to Z1 and Z2. Let's assume these are A1 and A2, respectively.

The percentage of children scoring between 25 and 28.5 can be calculated as:

Percentage = (A2 - A1) * 100

C) To calculate the number of children expected to have scores between 28 and 31.5, we need to calculate the area under the standard normal curve between the corresponding z-scores.

Z1 = (28 - 30.9) / 2.08
Z2 = (31.5 - 30.9) / 2.08

Using the z-table or a statistical calculator, find the areas under the standard normal curve corresponding to Z1 and Z2. Let's assume these are A1 and A2, respectively.

Expected number of children = (A2 - A1) * n
where n is the sample size, given as 20 in this case.

D) To find the raw score cutoff for the top 25% of children, we need to calculate the z-score corresponding to the top 25th percentile and then use the rearranged formula to find the raw score.

Z-score can be found for the 75th percentile using the z-table or a statistical calculator.

Formula:
X = Z * σ + μ

Plug in the values into the formula to find the raw score cutoff.