Chapter 18, Problem 38E

(a)

To determine

To find out during what percentage of years does Ithaca get more than $40\text{'}\text{'}$ of rain.

Answer to Problem 38E

  $13.7\%$ .

Explanation of Solution

It is given in the question that Ithaca, NY, gets an average of $35.4\text{'}\text{'}$ of rain each year, with the standard deviation of $4.2\text{'}\text{'}$ . And assume that a Normal model applies. Thus, let us find the z -score as:

  $\begin{array}{l}{z}_{40}=\frac{40-35.4}{4.2}\\ =1.095\end{array}$

Now, using the graphical utility to find the percentage of years does Ithaca get more than $40\text{'}\text{'}$ of rain as:

  $\begin{array}{l}P(z>1.095)=normalcdf(1.095,E99,0,1)\\ =0.137\end{array}$

Thus, the percentage of years does Ithaca get more than $40\text{'}\text{'}$ of rain is $13.7\%$ .

(b)

To determine

To find out less than how much rain falls in the driest $20\%$ of all years.

Answer to Problem 38E

  $31.865$ inches.

Explanation of Solution

It is given in the question that Ithaca, NY, gets an average of $35.4\text{'}\text{'}$ of rain each year, with the standard deviation of $4.2\text{'}\text{'}$ . And assume that a Normal model applies. Thus, using the graphical calculator to find the $invNorm(0.20,0,1)$ to find the z -score of $-0.842$ . Thus, plug in z -score to find the value of x as:

  $\begin{array}{l}-0.842=\frac{x-35.4}{4.2}\\ \Rightarrow -0.842\times 4.2=x-35.4\\ \Rightarrow x=35.4-0.842\times 4.2\\ \Rightarrow x=31.865\end{array}$

Thus, $31.865$ inches rain falls in the driest $20\%$ of all years.

(c)

To determine

To describe the sampling distribution model of this sample mean $\overline{y}$ .

Explanation of Solution

It is given in the question that Ithaca, NY, gets an average of $35.4\text{'}\text{'}$ of rain each year, with the standard deviation of $4.2\text{'}\text{'}$ . And assume that a Normal model applies. Thus, the conditions are as:

Random condition: It is satisfied as we are assuming that years are representative.

  $10\%$ condition: The sample size less than $10\%$ of the population size.

Thus, the conditions are met. And the model of this sampling distribution is the Normal model and the mean and the standard deviation is as follows:

  $\begin{array}{l}\mu ={\mu }_{\overline{y}}=35.4\\ {\sigma }_{\overline{y}}=\frac{\sigma }{\sqrt{n}}\\ =\frac{4.2}{\sqrt{4}}\\ =2.1\end{array}$

(d)

To determine

To find out what is the probability that those $4$ years average less than $30\text{'}\text{'}$ of rain.

Answer to Problem 38E

  $0.005$ .

Explanation of Solution

It is given in the question that Ithaca, NY, gets an average of $35.4\text{'}\text{'}$ of rain each year, with the standard deviation of $4.2\text{'}\text{'}$ . And assume that a Normal model applies. Thus, let us find the z -score as:

  $\begin{array}{l}{z}_{30}=\frac{30-35.4}{4.2}\\ =-2.571\end{array}$

Thus, using graphing utility to find the probability that those $4$ years average less than $30\text{'}\text{'}$ of rain is calculated as:

  $\begin{array}{l}P(z<-2.571)=normalcdf(-E99,-2.571,0,1)\\ =0.005\end{array}$