Question

Dementia is the loss of the in actual and social abilities severe enough to interfere with judging behavior and daily functioning. Alzheimer's disease is the most common type of dementia. In the article "Living with Early Onsite dementia: Exploring the Experience and Developing Evidence Guidelines for Practice", P Harris and J Keady explored the experiment struggles of people diagnosed with dementia and their familiar simple random sample \(21\) people with early-onset dementia the following data on age at diagnosis in years.

At the \(1%\) significance level, do the data provide sufficient evidence to conclude that the mean age at diagnosis of all people with early onset dementia is less than \(55\) years old? Assume that the population is standard deviation is \(6.8\) years.

Short answer

The null hypothesis is not rejected and it is reasonable to believe that the mean age at diagnosis of all people with early dementia is less than \(55\) year old.

Step-by-step solution

Step 1. Given information

The standard deviation is \(68\), the mean is \(52.5\), the sample size is \(21\) and the level of significance is \(0.01\).

Step 2. Calculation

The hypothesis are,

\(H_{0}:\mu =55\)

\(H_{0}:\mu <55\)

The test statistics is,

\(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\)

Substitute the given values in above equation.

\(z=\frac{52.5-55}{\frac{6.8}{\sqrt{21}}}\)

\(=-1.68\)

The critical value for level of significance of \(0.01\) is \(-2.33\)

The graph is shown below.

Since, \(z>z_{0.025}\)

Thus, the null hypothesis is not rejected and it is reasonable to believe that the mean age at diagnosis of all people with early dementia is less than \(55\) year old.