Age at onset of dementia was determined for a sample pf adults between the ages of 60 and 75. For 15 subjects, the results were SumX=1008, and Sum(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?
My bad. Thanks.
To answer these questions, we need to perform some calculations based on the given information:
a. Mean and SD calculation:
The mean (π₯Μ) can be determined by dividing the sum of the observations (SumX) by the total number of subjects (n). In this case, π₯Μ = SumX / n = 1008 / 15 = 67.2.
To calculate the standard deviation (SD), we need the sum of squares of the differences between each observation (X) and the mean (M), which is given as Sum(X-M)^2 = 140.4. SD can be found by taking the square root of the sum of squares divided by the total number of subjects minus 1 (n - 1). So, SD = sqrt(Sum(X-M)^2 / (n - 1)) = sqrt(140.4 / (15 - 1)) = sqrt(140.4 / 14) β 3.08.
Therefore, the mean is 67.2, and the standard deviation is approximately 3.08.
b. Calculating the percentage of people showing signs at or before age 62:
To determine the percentage, we need to convert the age of 62 to a z-score. The z-score formula is given as Z = (X - π₯Μ) / SD, where X is the value (age) in question, π₯Μ is the mean, and SD is the standard deviation.
Z = (62 - 67.2) / 3.08 β -1.69.
We can then use the Normal Curve Tables (also known as Z-tables) to find the percentage. From the Z-tables, we can find that the area to the left of -1.69 is approximately 0.0446.
To find the percentage of people who might start to show signs of dementia at or before age 62, we subtract this value from 1 and multiply by 100: (1 - 0.0446) * 100 β 95.54%
Therefore, approximately 95.54% of people might show signs of dementia at or before age 62 based on the given data and the Normal Curve Tables.
c. Determining the ages corresponding to Β±1 z-score from the mean:
To find the ages corresponding to Β±1 z-score from the mean, we can use the z-score formula:
Z = (X - π₯Μ) / SD.
For Z = -1 (1 z-score below the mean):
-1 = (X - 67.2) / 3.08.
Rearranging the equation, we find:
X - 67.2 = -3.08,
X β 64.12.
For Z = +1 (1 z-score above the mean):
1 = (X - 67.2) / 3.08.
Rearranging the equation, we find:
X - 67.2 = 3.08,
X β 70.28.
Therefore, the ages corresponding to Β±1 z-score from the mean are approximately 64.12 and 70.28.
d. Finding the ages marking the boundaries for the top 10% and bottom 10%:
The top and bottom 10% boundaries correspond to z-scores of Β±1.28, respectively (since 10% in each tail gives a total of 20%).
For the top boundary:
1.28 = (X - 67.2) / 3.08.
X - 67.2 = 3.95.
X β 71.15.
For the bottom boundary:
-1.28 = (X - 67.2) / 3.08.
X - 67.2 = -3.95.
X β 63.25.
Therefore, the ages marking the boundaries for the top 10% and bottom 10% of the distribution are approximately 71.15 and 63.25, respectively.