Question

The mean number of minutes for app engagement by a tablet use in $8.2$minutes. Suppose the standard deviation is one minute . Take a sample size of $70$.

a. What is probability of that the sum of the sample is between seven hours and ten hours? What does this mean in context of the problem ?

b. Find the ${84}^{th}$and ${16}^{th}$percentiles for the sum of the sample. interprets these values in context.

Short answer

The value of$a=2$

Step-by-step solution

Part (a) Step 1: Given Information

We have given that the app engagement by a tablet use in $8.2$minutes.

we need to find the probability of that the sample is between seven hours and ten hours and ${84}^{th}$and${16}^{th}$percentiles.

Part (a) Step 2: Simplify

We must find the sum of the sample between $7h\left(420min\right)$and $10h\left(600min\right)$

$P(420<\sum _{}^{}x<600)=$normalcdf $(420,600,574,8.37)=0.9991$

Or

$$\begin{gathered} Pr(420\sum _{}^{}x<600)=Pr\left(\frac{420-574}{8.37}<Z<\frac{600-574}{8.37}\right) \\ =Pr(-18.399<Z3.1063) \\ =Pr(Z<3.1063)-Pr(Z<-18.399) \\ =0.991-0 \\ =0.9991 \end{gathered}$$

Normal distribution:$Pr(420<x<600)=0.9991$

Part (b) Step 3: Calculation

Let $k_1=$the ${84}^{th}$percentile.

$k_1=invNorm(0.84,574,8.37)=582.3236$

Let $k_2=$the ${16}^{th}$percentile.

$K_2=invNorm(0.16,574,8.37)=565.6764$