Question

in new york city, the cost of a taxi ride is based on the distance for the trip. the cost includes a starting fee of $5.00 plus $3.25 for each additional mile. the cost for a taxi ride can be determined with this equation: cost = 5 + 3.25(distance)
danielle keeps track of all her taxi rides for one year. here are the summary statistics for the number of miles travelled during these trips.

n	mean	sd	min	q1	med	q3	max
12	6.75	6.61	1	2.5	3.5	10.5	23

a. danielles longest taxi ride was 23 miles to the museum of modern art. use the 1.5 x iqr rule to determine if this is an outlier. show your work.
b. what was the cost for danielles taxi ride to the museum of modern art?
c. suppose we found the cost for each of danielles taxi rides. what is the q1 and q3 for the cost of the taxi rides?
d. use the 1.5 x iqr rule to determine if the cost for danielles taxi ride to the museum of modern art is an outlier. show your work.

1. the distance, in miles, traveled to work for each person in a sample is recorded. after the data are collected, all the distances are converted to kilometers by multiplying each value by 1.609. which of the following statistics will remain the same for both units, miles and kilometers?

a) the median of the distances
b) the z -scores of the distances
c) the mean of the distances
d) the range of the distances
e) the minimum of the distances

Explanation:

Step1: Calculate the IQR for miles

The inter - quartile range (IQR) is calculated as $IQR = Q_3 - Q_1$. Given $Q_1 = 2.5$ and $Q_3=10.5$, so $IQR=10.5 - 2.5=8$.

Step2: Determine the boundaries for outliers for miles

The lower boundary for outliers is $Q_1-1.5\times IQR=2.5-1.5\times8=2.5 - 12=- 9.5$. The upper boundary for outliers is $Q_3 + 1.5\times IQR=10.5+1.5\times8=10.5 + 12 = 22.5$. Since $23>22.5$, the 23 - mile ride is an outlier for miles.

Step3: Calculate the cost of the 23 - mile ride

The cost formula is $cost = 5+3.25\times distance$. Substitute $distance = 23$ into the formula: $cost=5 + 3.25\times23=5+74.75 = 79.75$.

Step4: Calculate $Q_1$ and $Q_3$ for cost

We know that $cost = 5+3.25\times distance$. For $Q_1$ of distance ($Q_{1d}=2.5$), $Q_{1c}=5+3.25\times2.5=5 + 8.125=13.125$. For $Q_3$ of distance ($Q_{3d}=10.5$), $Q_{3c}=5+3.25\times10.5=5+34.125 = 39.125$.

Step5: Calculate the IQR for cost

$IQR_c=Q_{3c}-Q_{1c}=39.125 - 13.125 = 26$.

Step6: Determine the boundaries for outliers for cost

The lower boundary for outliers for cost is $Q_{1c}-1.5\times IQR_c=13.125-1.5\times26=13.125 - 39=-25.875$. The upper boundary for outliers for cost is $Q_{3c}+1.5\times IQR_c=39.125+1.5\times26=39.125 + 39 = 78.125$. Since $79.75>78.125$, the cost of the 23 - mile ride is an outlier for cost.

Step7: Analyze the effect of unit conversion on statistics

The $z - score$ is calculated as $z=\frac{x-\mu}{\sigma}$. When we convert from miles to kilometers by multiplying each data - point $x$ by a constant $k = 1.609$, the mean $\mu$ and the standard deviation $\sigma$ are also multiplied by $k$. So, $z_{new}=\frac{kx - k\mu}{k\sigma}=\frac{x-\mu}{\sigma}=z_{old}$. The median, mean, range and minimum will all be multiplied by 1.609.

Answer:

a. The 23 - mile ride is an outlier.
b. The cost is $\$79.75$.
c. $Q_1$ for cost is $\$13.125$ and $Q_3$ for cost is $\$39.125$.
d. The cost of the 23 - mile ride is an outlier.

1. B. The $z$-scores of the distances