Exercise 6-8
The Downtown Parking Authority of Tampa, Florida, reported the following information for a sample of 235 customers on the number of hours cars are parked and the amount they are charged.
Number of Hours Frequency Amount Charged
1 18 $3
2 33 7
3 49 10
4 44 15
5 34 19
6 15 24
7 8 27
8 34 30
235
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(a) Convert the information on the number of hours parked to a probability distribution. (Round your answers to 3 decimal places.)
Hours Probability
1
2
3
4
5
6
7
8
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(b-1) Find the mean and the standard deviation of the number of hours parked. (Round your intermediate values and final answers to 3 decimal places.)
Mean
Standard deviation
(b-2) How long is a typical customer parked? (Round your answer to 3 decimal places.)
The typical customer is parked for hours
(c) Find the mean and the standard deviation of the amount charged. (Round your intermediate values and final answers to 3 decimal places.)
Mean
Standard deviation
In (a) how do I convert the number of hours to a probability distribution
To convert the information on the number of hours parked to a probability distribution, we need to calculate the probability of each number of hours based on the given frequency.
(a) To calculate the probability, divide each frequency by the total number of customers.
Hours Frequency Probability
1 18 18/235 = 0.077
2 33 33/235 = 0.140
3 49 49/235 = 0.209
4 44 44/235 = 0.187
5 34 34/235 = 0.145
6 15 15/235 = 0.064
7 8 8/235 = 0.034
8 34 34/235 = 0.145
Therefore, the probability distribution for the number of hours parked is:
Hours Probability
1 0.077
2 0.140
3 0.209
4 0.187
5 0.145
6 0.064
7 0.034
8 0.145
(b-1) To find the mean and standard deviation of the number of hours parked, we can calculate the weighted average.
Mean = (Hours * Probability) for all hours
= (1*0.077) + (2*0.140) + (3*0.209) + (4*0.187) + (5*0.145) + (6*0.064) + (7*0.034) + (8*0.145)
= 3.692 (rounded to 3 decimal places)
To find the standard deviation, we use the formula:
Standard deviation = sqrt( ( (Hours - Mean)^2 * Probability ) for all hours )
= sqrt( ( (1-3.692)^2 * 0.077) + ( (2-3.692)^2 * 0.140) + ( (3-3.692)^2 * 0.209) + ( (4-3.692)^2 * 0.187) + ( (5-3.692)^2 * 0.145) + ( (6-3.692)^2 * 0.064) + ( (7-3.692)^2 * 0.034) + ( (8-3.692)^2 * 0.145) )
= 1.811 (rounded to 3 decimal places)
Therefore, the mean of the number of hours parked is 3.692 and the standard deviation is 1.811.
(b-2) To find how long a typical customer is parked, we can use the mean value calculated in part (b-1). So a typical customer is parked for approximately 3.692 hours.
(c) To find the mean and standard deviation of the amount charged, we can perform the same calculations as in part (b-1) and (b-2), but this time using the amount charged instead of the number of hours.
Mean = (Amount Charged * Probability) for all amounts
= (3*0.077) + (7*0.140) + (10*0.209) + (15*0.187) + (19*0.145) + (24*0.064) + (27*0.034) + (30*0.145)
= 13.926 (rounded to 3 decimal places)
Standard deviation = sqrt( ( (Amount Charged - Mean)^2 * Probability ) for all amounts )
= sqrt( ( (3-13.926)^2 * 0.077) + ( (7-13.926)^2 * 0.140) + ( (10-13.926)^2 * 0.209) + ( (15-13.926)^2 * 0.187) + ( (19-13.926)^2 * 0.145) + ( (24-13.926)^2 * 0.064) + ( (27-13.926)^2 * 0.034) + ( (30-13.926)^2 * 0.145) )
= 7.714 (rounded to 3 decimal places)
Therefore, the mean amount charged is 13.926 and the standard deviation is 7.714.
(a) To convert the information on the number of hours parked to a probability distribution, we need to divide the frequency of each hour by the total sample size.
Hours Frequency Probability
1 18 18/235 = 0.077
2 33 33/235 = 0.140
3 49 49/235 = 0.209
4 44 44/235 = 0.187
5 34 34/235 = 0.145
6 15 15/235 = 0.064
7 8 8/235 = 0.034
8 34 34/235 = 0.145
(b-1) To find the mean and standard deviation of the number of hours parked, we need to calculate the weighted average and the squared deviation from the mean.
Mean = (1*0.077) + (2*0.140) + (3*0.209) + (4*0.187) + (5*0.145) + (6*0.064) + (7*0.034) + (8*0.145) ≈ 3.891
Standard deviation = sqrt[(1-3.891)^2*0.077 + (2-3.891)^2*0.140 + (3-3.891)^2*0.209 + (4-3.891)^2*0.187 + (5-3.891)^2*0.145 + (6-3.891)^2*0.064 + (7-3.891)^2*0.034 + (8-3.891)^2*0.145] ≈ 1.848
(b-2) A typical customer is parked for the mean number of hours, which is approximately 3.891 hours.
(c) To find the mean and standard deviation of the amount charged, we need to calculate the weighted average and the squared deviation from the mean.
Mean = (3*0.077) + (7*0.140) + (10*0.209) + (15*0.187) + (19*0.145) + (24*0.064) + (27*0.034) + (30*0.145) ≈ $11.371
Standard deviation = sqrt[(3-11.371)^2*0.077 + (7-11.371)^2*0.140 + (10-11.371)^2*0.209 + (15-11.371)^2*0.187 + (19-11.371)^2*0.145 + (24-11.371)^2*0.064 + (27-11.371)^2*0.034 + (30-11.371)^2*0.145] ≈ $9.233