Question

In a recent study reported Oct.29, 2012 on the Flurry Blog, the mean age of tablet users is 35 years. Suppose the standard deviation is ten years. The sample size is 39 .

a. What are the mean and standard deviation for the sum of the ages of tablet users? What is the distribution?

b. Find the probability that the sum of the ages is between 1,400 and 1,500 years.

c. Find the ${90}^{\text{th}}$ percentile for the sum of the 39 ages.

Short answer

(a) The central limit theorem proves that the distribution is normal for sums.

$$\begin{gathered} \mu \sum _{ix}=1365\text{years} \\ {}^{\sigma }\sum x=62.4 \end{gathered}$$

(b) The required value is

$P\left(1400<\sum X<1500\right)=0.2723$

(c) The resultant value is $k=1445\text{years.}$

Step-by-step solution

Part (a) Step 1: Given information

Given,

The mean age of tablet users is $35$years

The value of Standard deviation is $10$years

The Sample size is$39$.

Part (a) Step 2: Simplification

The mean and standard deviation will follow a normal distribution, the distribution of tablet users' ages will be as follows:

$$\begin{gathered} \sum X-N\left(n{\mu }_X,\sqrt{n}{\sigma }_X\right) \\ \sum X~N(39\times 35,\sqrt{39}\times 10) \\ \sum X~N(1365,62.44) \end{gathered}$$

Part (b) Step 1: Given information

Given,

The mean age of tablet users $=35$ years

Standard deviation is ten years

Sample size is 39

Part (b) Step 2: Simplification

Use a Ti-83 calculator to calculate the chance that the sum of ages is between 1400 and 1500 years. To do so, go to $2^{nd}$then DISTR, then scroll down to the normcdf option and fill in the required information. After that, press the calculator's ENTER key to get the desired result. The screenshot is as follows:

The chance that the total age falls between 1400 and 1500 years is $0.2722$.

Part (c) Step 1: Given information

Given,

The mean age of tablet users $=35$years

The Standard deviation is ten years

Sample size is 39

Part (c) Step 2: Simplification

To calculate ${90}^{th}$percentile for the sums, use Ti-83 calculator. For this, click on $2^{nd}$, then DISTR and then scroll down to the invnorm option and enter the provided details. After this, click on ENTER button of calculator to have the desired result. The screenshot is given as below:

The quartile of the${90}^{th}$ percentile for the sums of 39 ages is $1445.02$.