Chapter 7, Problem R7.7RE

(a)

To determine

To Explain: the reasonable to use a normal distribution to approximate the sampling distribution of $\overline{x}$ for SRSs of size 50.

Answer to Problem R7.7RE

Sample is sufficiently large and therefore it can apply the central limit theorem

Explanation of Solution

The central limit theorems say that the sampling distribution of the sample means is about normal, when the sample is sufficiently large. In general, it is consider a sample to be sufficiently large, when the sample size is 10 or more.

In this case, the sample size is 50, which is more than 30 and therefore the sampling distribution of the sample means $\overline{x}$ is about normal, by the central limit theorem.

(b)

To determine

To Calculate: the probability that the mean number o months in a sample of size 50 are greater than or equal to 0.6

Answer to Problem R7.7RE

0.1562

Explanation of Solution

Given:

  $\begin{array}{c}\mu =0.5\\ \sigma =0.7\\ n=50\end{array}$

Formula used:

  $\begin{array}{c}{\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\\ z=\frac{x-\mu }{\sigma }\end{array}$

Calculation:

The mean of the sampling distribution of the sample mean $\overline{x}$ is equal to the population mean:

  ${\mu }_{\overline{x}}=\mu =0.5$

The standard deviation of the sampling distribution of the sample mean $\overline{x}$ is

  $\begin{array}{c}{\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\\ =\frac{0.7}{\sqrt{50}}\\ =0.0990\end{array}$

The sample size is 30 or more. By, the central limit theorem, we then know that the sampling distribution of the sample mean $\overline{x}$ is approximately normal.

The z-score is

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ =\frac{0.6-0.5}{0.0990}\\ =1.01\end{array}$

The associative probability using table

  $\begin{array}{c}P\left(\overline{x}\ge 0.6\right)=P\left(z>1.01\right)\\ =P\left(Z<-1.01\right)\\ =0.1562\end{array}$

(c)

To determine

To Explain: that there is convincing evidence that the month population is getting larger in this state.

Answer to Problem R7.7RE

No

Explanation of Solution

No

Because this probability is not small, it is plausible that the sample mean number of months is this high by chance alone.