Question

Please help me i don’t know how to do this

Answer 1

The question provides the following information:

`\begin{gathered} mean=\mu=2.8 \\ standard\text{ }deviation=\sigma=1.5 \end{gathered}`

QUESTION A

The distribution of a normal variable is given in the form:

`X\sim N(\mu,\sigma^2)`

Substituting the values, we have:

`\begin{gathered} X\sim N(2.8,1.5^2) \\ \Rightarrow X\sim N(2.8,2.25) \end{gathered}`

QUESTION B

We will use the z-score to calculate the probability. The formula is given to be:

`z=(x-\mu)/(\sigma)`

Take x = 3.1. Therefore, we have:

`z=(3.1-2.8)/(1.5)=0.2`

The probability will be calculated to be:

`\begin{gathered} P(Z\leq0.2)=Pr(Z\leq0)+Pr(0\leq Z\leq0.2) \\ \therefore \\ P(Z\leq0.2)=0.5+0.0793 \\ P(Z\leq0.2)=0.5793 \end{gathered}`

The probability is 0.5793.

QUESTION C

The z-score for 3.2 hours is calculated to be:

`z=(3.2-2.8)/(1.5)=0.27`

The probability is calculated to be:

`\begin{gathered} P(Z\ge0.27)=Pr(Z\ge0)-Pr(0\leqslant Z\leqslant0.27) \\ \therefore \\ P(Z\ge0.27)=0.5-0.1064 \\ P(Z\ge0.27)=0.3936 \end{gathered}`

In percentage, the answer will be 39.36%.

QUESTION D

77% of the children will have a probability of 0.77. The z-score corresponding to this will be:

`z=0.7388`

Therefore, we can use the formula for the z-score to get the corresponding value of x:

`\begin{gathered} 0.7388=(x-2.8)/(1.5) \\ x-2.8=0.7388*1.5=1.1082 \\ x=2.8+1.1082 \\ x=3.9082 \end{gathered}`

The number of hours is 3.9082 hours, to four decimal places.