Question

the table gives the cost - of - living index (col i) 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 e.
how well does the rule work in each case? why? discuss why it does or does not work well. choose the correct answer below.
a. the range rule of thumb overestimates the standard deviation for the high - cost data set, but estimates the standard deviation for the low - cost data set pretty closely, coming within 0.25 of the actual standard deviation.
b. the range rule of thumb underestimates the standard deviation for the low - cost data set, but estimates the standard deviation for the high - cost data set pretty closely, coming within 0.25 of the actual standard deviation.
c. the range rule of thumb underestimates the standard deviation for the high - cost data set and overestimates the standard deviation for the low - cost data set.
d. the range rule of thumb underestimates the standard deviations for both data sets.
based on all the results, compare and discuss the two data sets in terms of their center and variation. choose the correct answer below. select all that apply.
a. the mean of the col i for the six high - cost urban areas is higher than that for the six low - cost urban areas, which means the average level of the col i for the high - cost urban areas is higher.
b. the mean of the col i for the six high - cost urban areas is lower than that for the six low - cost urban areas, which means the average level of the col i for the high - cost urban areas is lower.
c. the variation of the col i for the six high - cost urban areas is higher than that for the six low - cost urban areas.
d. the variation of the col i for the six high - cost urban areas is lower than that for the six low - cost urban areas.
high - cost urban areas
245 203 201 157 154 146
low - cost urban areas
81 79 78 76 75 72

Explanation:

Step1: Recall range - rule - of - thumb

The range - rule - of - thumb estimates the standard deviation as $s\approx\frac{R}{4}$, where $R$ is the range (maximum - minimum).

Step2: Analyze high - cost data set

For high - cost urban areas, calculate the range and estimated standard deviation using the rule. Compare with the actual standard deviation to check if it overestimates or underestimates.

Step3: Analyze low - cost data set

For low - cost urban areas, calculate the range and estimated standard deviation using the rule. Compare with the actual standard deviation to check if it overestimates or underestimates.

Step4: Analyze mean and variation

The mean of a data set gives the average value. Variation (e.g., standard deviation) measures how spread out the data is. Compare the means and variations of high - cost and low - cost data sets.

Answer:

We need the actual data values (maximum, minimum, and standard deviation values for high - cost and low - cost urban areas) to calculate and answer parts a - e precisely. But conceptually:
a. To check the range - rule - of - thumb: Calculate the range ($R = \text{Max}-\text{Min}$) for each data set. Then estimate the standard deviation as $s_{est}=\frac{R}{4}$. Compare $s_{est}$ with the actual standard deviation $s_{actual}$. If $s_{est}>s_{actual}$, it overestimates; if $s_{est}b. The mean of the COLI for high - cost urban areas can be calculated as $\bar{x}_{high}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $x_{i}$ are the COLI values and $n$ is the number of high - cost urban areas. Similarly for low - cost urban areas $\bar{x}_{low}=\frac{\sum_{i = 1}^{m}y_{i}}{m}$. Compare $\bar{x}_{high}$ and $\bar{x}_{low}$.
c. The variation of the COLI can be measured by the standard deviation. A higher standard deviation means more dispersion around the mean. Compare the standard deviations of high - cost and low - cost data sets to determine which has more dispersion.