Question

A small ferry runs every half hour from one side of a large river to the other. The probability distribution for the random variable = money collected (in dollars) on a randomly selected ferry trip is shown here. The mean of this probability distribution is $19.35.

Money collected
0
5
10
15
20
25


Probability
0.02
0.05
0.08
0.16
0.27
0.42



What is the standard deviation of ?
Interpret this value.

A) = 41.3275. If many, many ferry trips are randomly selected, the cost will be exactly $6.43 away from the mean of $19.35.

B) = 41.3275. If many, many ferry trips are randomly selected, the cost will typically vary by about $41.33 from the mean of $19.35.

C) = 6.429. If many, many ferry trips are randomly selected, the cost will be exactly $6.43 away from the mean of $19.35.

D) The standard deviation cannot be calculated in this situation because we do not have a list of randomly selected ferry trip fees.

E) = 6.429. If many, many ferry trips are randomly selected, the cost will typically vary by about $6.43 from the mean of $19.35.

Answer 1

The standard deviation of this probability distribution is 6.429 dollars. If many ferry trips are randomly selected, the cost will typically vary by about $6.43 from the mean of $19.35.

Step-by-step explanation:

The standard deviation of the probability distribution is calculated by finding the square root of the sum of the squared differences between each value of and the mean, weighted by their respective probabilities. In this case, the mean of is $19.35. The calculation is as follows:

1. Subtract the mean from each value:

• 0 - 19.35 = -19.35
• 5 - 19.35 = -14.35
• 10 - 19.35 = -9.35
• 15 - 19.35 = -4.35
• 20 - 19.35 = 0.65
• 25 - 19.35 = 5.65

Square each difference:

• (-19.35)^2 = 374.9225
• (-14.35)^2 = 205.8225
• (-9.35)^2 = 87.4225
• (-4.35)^2 = 18.9225
• (0.65)^2 = 0.4225
• (5.65)^2 = 31.9225

Multiply each squared difference by its respective probability:

• 0.02 * 374.9225 = 7.49845
• 0.05 * 205.8225 = 10.291125
• 0.08 * 87.4225 = 6.9538
• 0.16 * 18.9225 = 3.0276
• 0.27 * 0.4225 = 0.114075
• 0.42 * 31.9225 = 13.39905

Sum up the products:

• 7.49845 + 10.291125 + 6.9538 + 3.0276 + 0.114075 + 13.39905 = 41.2831

Take the square root of the sum:

• sqrt(41.2831) = 6.429

Therefore, the standard deviation of is 6.429 dollars. If many, many ferry trips are randomly selected, the cost will typically vary by about $6.43 from the mean of $19.35.