Question

when sam goes to a restaurant, he always tips the server $2 plus 10% of the cost of the meal. if sams distribution of meal costs has a mean of $9 and a standard deviation of $3, what are the mean and standard deviation of his tip distribution?
$2.90, $2.30
$2.90, $0.30
$9.00, $3.00
$11.00, $2.00

Explanation:

Step1: Find the mean of the tip distribution

Let \(x\) be the cost of the meal. The tip \(y = 2+0.1x\).
The mean of \(x\) is \(\mu_x = 9\).
Using the formula for the mean of a linear transformation \(y=a + bx\) (\(a = 2\), \(b=0.1\)), \(\mu_y=a + b\mu_x\).
Substitute \(a = 2\), \(b = 0.1\), \(\mu_x=9\) into the formula: \(\mu_y=2+0.1\times9=2 + 0.9=\$2.9\).

Step2: Find the standard deviation of the tip distribution

The standard deviation of \(x\) is \(\sigma_x = 3\).
Using the formula for the standard deviation of a linear transformation \(y=a+bx\) (\(a = 2\), \(b = 0.1\)), \(\sigma_y=\vert b\vert\sigma_x\).
Substitute \(b = 0.1\), \(\sigma_x = 3\) into the formula: \(\sigma_y=0.1\times3=\$0.3\).

Answer:

\(\$2.90,\$0.30\) (the second option)