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" (Al=hcimer's Care Quarterly, Vol. 5, Issue 2, pp. 111-122), P. Harris and J. Keady explored the experience and struggles of people diagnosed with dementia and their families. If the mean age at diagnosis of all people with early-onset dementia is 55 years, find the probability that a random sample of $21$ such people will have a mean age at diagnosis less than $52.5$ years. Assume that the population standard deviation is $6.8$ years. State any assumptions that you are making in solving this problem.

Short answer

Chance of a random sample of $21$ such patients having a mean age at diagnosis of less than $52.5$ years is $0.0465$

Step-by-step solution

Given information

All patients with early-onset dementia are on average $55$ years old when they are diagnosed. Assume the standard deviation of the population is $6.8$ years.

Calculation

We want to know how likely it is that a random sample of $21$ patients with this condition will have a mean age at diagnosis of less than $52.5$ years.

We assume that all persons with early-onset dementia have a diagnosis that is roughly normal distributed.

We have $\mu =55,\sigma =6.8$, and $n=21$ based on the above data.

Calculation

The probability that persons will be diagnosed at a younger age than $52.5$years.

$$\begin{gathered} P(\overline{X}<52.5)=P\left(\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}<\frac{52.5-55}{6.8/\sqrt{21}}\right) \\ =P\left(Z<\frac{-2.5}{1.483882}\right) \\ =P(Z<-1.68) \\ =0.0465 \end{gathered}$$

As a result, the chance of a random sample of $21$ such patients having a mean age at diagnosis of less than $52.5$ years is $0.0465$