Question

the table gives the cost of living index (coli) for six urban areas with a relatively high cost of living and six urban areas with a relatively low cost of living. the index is based on costs for housing, utilities, grocery items, transportation, health care, and miscellaneous goods and services. 100 represents the national average. complete parts a through d.
construct the box - plot for the low - cost urban areas. choose the correct graph below.
a
b
c
d
c. find the standard deviation for each of the two data sets.
the standard deviation for the high - cost urban areas is
(type an integer or decimal rounded to two decimal places as needed.)
the standard deviation for the low - cost urban areas is
(type an integer or decimal rounded to two decimal places as needed.)
d. apply the range rule of thumb to estimate the standard deviation of each of the two data sets. how well does the rule work in each case? briefly discuss why it does or does not work well.
the standard deviation for the high - cost urban areas is approximately
(type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Recall standard - deviation formula

The formula for the sample standard deviation is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$, where $x_{i}$ are the data - points, $\bar{x}$ is the mean, and $n$ is the number of data - points.

Step2: Calculate the mean for high - cost urban areas

High - cost urban areas data: $245,203,201,157,154,146$.
$\bar{x}_{1}=\frac{245 + 203+201+157+154+146}{6}=\frac{1106}{6}\approx184.33$.

Step3: Calculate $(x_{i}-\bar{x}_{1})^{2}$ for high - cost urban areas

$(245 - 184.33)^{2}=(60.67)^{2}=3681.8489$, $(203 - 184.33)^{2}=(18.67)^{2}=348.5689$, $(201 - 184.33)^{2}=(16.67)^{2}=277.8889$, $(157 - 184.33)^{2}=(- 27.33)^{2}=746.9289$, $(154 - 184.33)^{2}=(-30.33)^{2}=920.9089$, $(146 - 184.33)^{2}=(-38.33)^{2}=1469.1889$.
$\sum_{i = 1}^{6}(x_{i}-\bar{x}_{1})^{2}=3681.8489+348.5689+277.8889+746.9289+920.9089+1469.1889 = 7445.3334$.

Step4: Calculate the standard deviation for high - cost urban areas

$s_{1}=\sqrt{\frac{7445.3334}{6 - 1}}=\sqrt{\frac{7445.3334}{5}}=\sqrt{1489.06668}\approx38.59$.

Step5: Calculate the mean for low - cost urban areas

Low - cost urban areas data: $81,81,79,76,75,72$.
$\bar{x}_{2}=\frac{81 + 81+79+76+75+72}{6}=\frac{464}{6}\approx77.33$.

Step6: Calculate $(x_{i}-\bar{x}_{2})^{2}$ for low - cost urban areas

$(81 - 77.33)^{2}=(3.67)^{2}=13.4689$, $(81 - 77.33)^{2}=(3.67)^{2}=13.4689$, $(79 - 77.33)^{2}=(1.67)^{2}=2.7889$, $(76 - 77.33)^{2}=(-1.33)^{2}=1.7689$, $(75 - 77.33)^{2}=(-2.33)^{2}=5.4289$, $(72 - 77.33)^{2}=(-5.33)^{2}=28.4089$.
$\sum_{i = 1}^{6}(x_{i}-\bar{x}_{2})^{2}=13.4689+13.4689+2.7889+1.7689+5.4289+28.4089 = 65.3334$.

Step7: Calculate the standard deviation for low - cost urban areas

$s_{2}=\sqrt{\frac{65.3334}{6 - 1}}=\sqrt{\frac{65.3334}{5}}=\sqrt{13.06668}\approx3.61$.

Step8: Recall the range rule of thumb

The range rule of thumb states that $s\approx\frac{R}{4}$, where $R$ is the range ($R=\text{Max}-\text{Min}$).

Step9: Calculate the range and estimated standard deviation for high - cost urban areas

For high - cost urban areas, $R_{1}=245 - 146 = 99$, $s_{1\text{est}}\approx\frac{99}{4}=24.75$.

Step10: Calculate the range and estimated standard deviation for low - cost urban areas

For low - cost urban areas, $R_{2}=81 - 72 = 9$, $s_{2\text{est}}\approx\frac{9}{4}=2.25$.

Step11: Discuss the range rule of thumb

For high - cost urban areas, the actual standard deviation ($s_{1}\approx38.59$) and the estimated standard deviation ($s_{1\text{est}} = 24.75$) have a significant difference. The range rule of thumb does not work well because the data is not very symmetrically distributed.
For low - cost urban areas, the actual standard deviation ($s_{2}\approx3.61$) and the estimated standard deviation ($s_{2\text{est}} = 2.25$) also have a difference. The data may not be symmetric enough for the range rule of thumb to be highly accurate.

Answer:

The standard deviation for the high - cost urban areas is approximately $38.59$.
The standard deviation for the low - cost urban areas is approximately $3.61$.
The standard deviation for the high - cost urban areas using the range rule of thumb is approximately $24.75$.
The standard deviation for the low - cost urban areas using the range rule of thumb is approximately $2.25$.