Chapter 8.4, Problem 18E

Mortgage rates: Following are interest rates (annual percentage rates) for a 30-year fixed rate mortgage from a sample of lenders in Macon. Georgia on June 20. 2013. Is reasonable to assume that the population is approximately normal.

1. Find the sample standard deviation s.
2. Construct a 95% confidence interval for population standard deviation
3. Base on the confidence interval. Is it reasonable to believe that the population standard deviation is greater than 0.5? Explain.

To determine

The sample standard deviation $s$.

Answer to Problem 18E

The sample standard deviation is $0.189$.

Explanation of Solution

Given information:

Following are interest rates (annual percentage rates) for a $30$ -year fixed-rate mortgage from a sample of lenders in Atlanta, Georgia, on August $9,2010$.

$4,327$	$4461$	$4.547$	$4.398$	$4.199$	$4.460$
$4.531$	$4.365$	$4.407$	$4.639$	$4.804$	$4.842$

It is assumed that the population is normally distributed.

Calculation:

$4,327$	$4461$	$4.547$	$4.398$	$4.199$	$4.460$
$4.531$	$4.365$	$4.407$	$4.639$	$4.804$	$4.842$

By using MINITAB sample standard deviation is calculated with the help of following steps :

1) Import the data.

2) Select the Stat and choose the Basic Statistics option.

3) Select the $1$ Variance and enter Summarized data.

4) Select Confidence level

5) Click ok.

Test and $\text{Cl}$

for One Variance: Rate

Statistics

$\begin{array}{l}\text{ Variable N StDev Variance}\\ \text{Rate 12 0}\text{.189 0}\text{.0357}\end{array}$

From the MIN TAB output, the sample standard deviation is $0.189$.

To determine

The $95\%$ confidence interval for the population standard deviation $\sigma$.

Answer to Problem 18E

A $95\%$ confidence interval for population standard deviation is $0.134\text{ to 0}\text{.321}$.

Explanation of Solution

Given information:

Following are interest rates (annual percentage rates) for a $30$ -year fixed-rate mortgage from a sample of lenders in Atlanta, Georgia, on August $9,2010$.

$4,327$	$4461$	$4.547$	$4.398$	$4.199$	$4.460$
$4.531$	$4.365$	$4.407$	$4.639$	$4.804$	$4.842$

It is assumed that the population is normally distributed.

Calculation:

By using MINITAB a $95\%$

confidence interval is calculated with the help of following steps :

1) Import the data.

2) Select the Stat and choose the Basic Statistics option.

3) Select the $1$ Variance and enter Summarized data.

4) Select Confidence level

5) Click ok.

Test and $\text{Cl}$

for One Variance: Rate

Statistics

$\begin{array}{l}\text{ Variable N StDev Variance}\\ \text{Rate 12 0}\text{.189 0}\text{.0357}\end{array}$

$95\%$ Confidence Intervals

$\begin{array}{l}\text{Variable Method }\text{CI for StDev CI for variance}\\ \text{Rate Chi-Square }\left(0.134,0.321\right)\left(0.0179,0.1029\right)\end{array}$

From the MINITAB output, a $95\%$ confidence interval for population standard deviation is $0.134\text{ to 0}\text{.321}$.

To determine

Whether it is reasonable to believe that the population standarddeviation is greater than $0.5$.

Answer to Problem 18E

It is not reasonable to believe that the population standard deviation is greater than $0.5$.

Explanation of Solution

Given information:

Following are interest rates (annual percentage rates) for a $30$ -year fixed-rate mortgage from a sample of lenders in Atlanta, Georgia, on August $9,2010$.

$4,327$	$4461$	$4.547$	$4.398$	$4.199$	$4.460$
$4.531$	$4.365$	$4.407$	$4.639$	$4.804$	$4.842$

It is assumed that the population is normally distributed.

Formula used:

Based on confidence interval, ¡t is reasonable to believe that the population standard deviation higher than $0.50$ due to a $95\%$ confidence interval for population standard deviation is between $\text{0}\text{.134}$ and $\text{0}\text{.321}$ does not include $0.500$.

Calculation:

From part $b$ of this exercise, a $95\%$ confidence interval for population standard deviation is $0.134\text{ to 0}\text{.321}$.

Since the $95\%$ confidence interval of population standard deviation $\sigma$, $0.134\text{ to 0}\text{.321}$, does not include $0.500$. This means there is $95\%$ chance that the population standard deviation is between $0.134\text{ to 0}\text{.321}$.

Hence it is not reasonable to believe that the population standard deviation is greater than $0.5$.