Q. 3.134 - Elementary Statistics

Question

 Early-Onset Dementia. Dementia is the loss of intellectual and social abilities severe enough to interfere with judgment, behavior, and daily functioning. Alzheimer's disease is the most common type of dementia. In the article "Living with Early Onset Dementia: Exploring the Experience and Developing Evidence-Based Guidelines for Practice" (Alsheimer's Care Quarterly), P. Harris and J. Keady explored the experience and struggles of people diagnosed with dementia and their families. The ages of a simple random sample of $21$ people with early-onset dementia have a mean of $52.5$ years and a standard deviation of $6.5$ years.

a. construct a graph
b. Apply Chebyshev's rule with $k=2$ to make pertinent statements about the observations in the sample.
c. repeat part (b) with $k=3$

Step-by-Step Solution

Answer

Part(a) Required figure is given below

Part(b) At least $16$ people will have early-onset dementia between $39.5$ and $65.5$ years.
Part(c) At least $19$ people will have early-onset dementia between $33$ and $72$ years.

Part(a) Step 1 : Given information

We are given that sample has $21$people.

Mean $\overline{x}=52.5$ 

Standard deviation $s=6.5$

Part(a) Step 2 : Simplify

According to Chebyshev's Rule,

$$\begin{gathered} \overline{x}=52.5 \\ \overline{x}-s=52.5-6.5=46.0 \\ \overline{x}-2s=52.5-2\times 6.5=39.5 \\ \overline{x}-3s=52.5-3\times 6.5=33.0 \\ \overline{x}+s=52.5+6.5=59.0 \\ \overline{x}+2s=52.5+2\times 6.5=65.5 \\ \overline{x}+3s=52.5+3\times 6.5=72.0 \end{gathered}$$

Required graph is

Part(b) Step 1 : Given information

We are given that sample has $21$people.

Mean $\overline{x}=52.5$

Standard deviation $s=6.5$

Part(b) Step 2 : Simplify

According to Chebyshev's rule ,if

$k=2$, then at least $75\%$ of the people in the sample have early onset dementia within two standard deviations to either side of the mean.

 Now,

$21\times 0.75\approx 16$

 Now, two standard deviations to either side of the mean is from 

$39.5$ to $65.5$ years.

Interpretation- At least $16$ out of $21$ people in the sample are with early-onset dementia between  $39.5$and $65.5$ years

Part(c) Step 1 : Given information

We are given that sample has $21$ people.

Mean $\overline{x}=52.5$

Standard deviation $s=6.5$

Part(c) Step 2 : Simplify

According to Chebyshev's rule ,if

 $k=3$, then at least $89\%$ of the people in the sample are with early onset dementia within two standard deviations to either side of the mean.

 Now,

$21\times 0.89\approx 19$

 Now, two standard deviations to either side of the mean is from 

$33$ to $72$ years.

Interpretation At least $19$ out of $21$ people in the sample are with early-onset dementia between $33$ and $72$ years