Question

in a recent year, taxicabs in los angeles charged an initial fee of $2.85 plus $2.70 per mile. in equation form, fare = 2.85 + 2.7 (miles). at the end of a month, a businessman collects all his taxicab receipts and calculates some numerical summaries. the mean fare he paid was $15.45 with a standard deviation of $10.20. what are the mean and standard deviation of the length of his cab rides in miles? mean = 5.722 miles, standard deviation = 3.778 miles mean = 4.667 miles, standard deviation = 3.778 miles mean = 4.667 miles, standard deviation = 2.722 miles mean = 5.722 miles, standard deviation = 2.722 miles

Explanation:

Step1: Find the mean of miles

We know that \(Fare = 2.85+2.7\times miles\). Let \(y\) be the fare and \(x\) be the miles. So \(y = 2.85 + 2.7x\).
The formula for the mean of a linear transformation \(y=a + bx\) is \(\mu_y=a + b\mu_x\). We are given \(\mu_y = 15.45\) and \(a = 2.85\), \(b = 2.7\).
We solve for \(\mu_x\):

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Step2: Find the standard deviation of miles

The formula for the standard deviation of a linear transformation \(y=a + bx\) is \(\sigma_y=\vert b\vert\sigma_x\). We are given \(\sigma_y = 10.20\) and \(b = 2.7\).
We solve for \(\sigma_x\):

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Answer:

Mean \( = 4.667\) miles, Standard deviation \(=3.778\) miles (the second option)