Question

Chebyshev's rule to make estimates.

Need help with b and c please.

Quiz Me: Quiz Me: 3.3-1 Use Chebyshev's rule to make estimates. Submit Quiz This Question: 1 pt 2 of 2 This Quiz: 2 pts possible Question Help 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 an article, two authors 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 52.8 years and a standard deviation of 4.7 years. Complete parts (a) through (c) below Type integers or decimals. Do not round.) b. Apply Chebyshev's rule with k 2 to make pertinent statements about the observations in the sample At least | out of Џ people in the sample have ages between L years and years Type integers or decimals. Do not round.) c. Apply Chebyshev's rule with k 3 to make pertinent statements about the observations in the sample At leastout of people in the sample have ages between years and years (Type integers or decimals. Do not round.) Enter your answer in each of the answer boxes

Answer 1

Here sample mean = 52.8 years

sample standard deviaiton = 4.7 years

$\bar x$ -3s = 52.8 - 3 * 4.7 = 38.7 years

$\bar x$ +3s = 52.8 + 3 * 4.7 = 66.9 years

$\bar x$ -2s = 52.8 - 2 * 4.7 = 43.4 years

$\bar x$ +2s = 52.8 + 2 * 4.7 = 62.2 years

$\bar x$ - s = 52.8 -  4.7 = 48.1 years

$\bar x$ +s = 52.8 + 4.7 = 57.5 years

$\bar x$ = 52.8

as we know by chebyshev's rule

Pr(l X - $\mu$ l $\geq$ k$\sigma$) $\leq$ 1/k²

so here as k = 2 the 1/k² = 1/4

so a total of 1 - 1/4 = 0.75 or 75% of people will come under values $\bar x$ -2s and $\bar x$ + 2s

so here expected number of people = 21 * 0.75 = 15.75

so here at least 15.75 out of 21 people in the sample have ages between 43.4 years and 62.2 years

so here as k = 3 the 1/k² = 1/9

so a total of 1 - 1/9 = 0.8889 or 88.89% of people will come under values $\bar x$ -3s and $\bar x$ + 3s

so here expected number of people = 21 * 0.8889 = 18.67

so here at least 18.67 out of 21 people in the sample have ages between 38.7 years and 66.9 years.