Question

Early-Onset Dementia. Dementia is the loss of the in tual and social abilities severe enough to interfere with judí behavior, and daily functioning. Alzheimer's disease is the moon type of dementia. In the article "Living with Early Onsmentia: Exploring the Experience and Developing Evidence Guidelines for Practice" (Alzheimer's Care Quarterly, Vol. 5, I pp. 111-122), P. Harris and J. Keady explored the experience struggles of people diagnosed with dementia and their family simple random sample of 21 people with early-onset dementia the following data on age at diagnosis, in years.

60	58	52	58	59	58	51
61	54	59	55	53	44	46
47	42	56	57	49	41	43

At the 1% significance level, do the data provide sufficient ev| to conclude that the mean age at diagnosis of all people with onset dementia is less than 55 years old? Assume that the point standard deviation is 6.8 years. (Note: $\mu$=52.5 years.)

Short answer

The presented data is adequate evidence to infer that the mean age of diagnosis of all patients with early-onset dementia is 55 years old at the 1% significance level.

Step-by-step solution

Step 1. Given information.

The given table is

60	58	52	58	59	58	51
61	54	59	55	53	44	46
47	42	56	57	49	41	43

Step 2. Let the mean age at diagnosis of all people with early-onset dementia be μ ,

Population standard deviation is,

$\sigma =6.8years$

Now, the hypotheses test is,

$$\begin{gathered} H_0:\mu =55years \\ {H}_a:\mu <55years \end{gathered}$$localid="1652289900230" $\begin{array}{r}{H}_0:\mu =55years\\ H_a:\mu <55years\end{array}$

We must conduct the test at a 1% level of significance, i.e.,$\alpha =0.01$

The sample size waslocalid="1652289909007" $n=21$

The sample mean is, $\overline{x}=52.5years$

Step 3. Now Test statistic, 

$$\begin{gathered} \mathrm{Test}\mathrm{statistic}:\mathrm{z}=\frac{\overline{\mathrm{x}}-{𝜇}_0}{{\displaystyle \frac{\mathrm{\sigma }}{\sqrt{\mathrm{n}}}}} \\ =\frac{52.5-55}{{\displaystyle \frac{6.8}{\sqrt{21}}}} \\ =-1.68 \end{gathered}$$

Step 4. Solution

Because the test is a two-tailed test, $\alpha =0.01$the critical value is$-Z_a=-Z_{0.01}$, localid="1652289915878" $-Z_a=-Z_{0.01}$.

Here, the region is,

localid="1652289920353" $z<-z_{0.01}$$z<-z_{0.01}$i.e.,

$z<-2.33$

Step 5. Here,

$z=-1.68>z_{0.01}=-2.33$

Because we do not reject at a 1% threshold of significance as the test statistic's value.

Furthermore, z does not lie within the rejection zone.

The presented data is adequate evidence to infer that the mean age of diagnosis of all patients with early-onset dementia is 55 years old at the 1% significance level.