Question

The Downtown Parking Authority of Tampa, Florida, reported the following information for a sample of 229 customers on the number of hours cars are parked and the amount they are charged.

Number of Hours Frequency Amount Charged 1 16 $ 3 2 34 6 3 51 12 4 39 16 5 34 21 6 16 24 7 9 27 8 30 29 229

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

a-2. Is this a discrete or a continuous probability distribution?

b-1. Find the mean and the standard deviation of the number of hours parked. (Do not round intermediate calculations. Round your final answers to 3 decimal places.)

b-2. How long is a typical customer parked? (Do not round intermediate calculations. Round your final answers to 3 decimal places.)

c. Find the mean and the standard deviation of the amount charged. (Do not round intermediate calculations. Round your final answers to 3 decimal places.)

Answer 1

Answer:

a

See in the explanation

a-2.

Discrete

b-1.

Mean = 4.201

Standard Deviation = 2.069

b-2.

4.201

c.

Mean = 16.153

Standard Deviation = 8.079

Step-by-step explanation:

Given Data:

Number of Hours         Frequency               Amount Charged  

          1                                 16                                $3        

          2                                34                                 6  

          3                                51                                  12  

          4                                39                                 16  

          5                                34                                 21

          6                                 16                                 24

          7                                  9                                  27

          8                                 30                                 29

                                       ∑f = 229

a. Convert the information on the number of hours parked to a probability distribution:

The probability is calculated by dividing each frequency by 229. For example probability of Hour 1 is calculated as:

16 / 229 = 0.06987

This way all the hours probabilities are computed. The probability distribution is given below

Hours          Probability

   1                0.06987

   2               0.14847

   3               0.2227

   4               0.1703

   5               0.1485

   6               0.0699

   7               0.0393

   8               0.1310

   ∑                    1

a-2. Is this a discrete or a continuous probability distribution?

This is a discrete probability distribution as the probability of each hour of between 0 and​ 1 and the sum of all the probabilities of hours is 1.

b-1. Find the mean and the standard deviation of the number of hours parked.

First multiply each value of Number of hours by the corresponding frequency. Let x represents the number of hours and f represents frequency. Then:

Number of Hours Parked

fx

16

68

153

156

170

96

63

240

Now add the above computed products.

∑fx = 16+68+153+156+170+96+63+240  = 962

Compute Mean:

Now the formula to calculate mean:

Mean = Sum of the value / Number of value

         = ∑fx / ∑f

         = 962 / 229

Mean = 4.201

Compute Standard Deviation:

Let x be the Number of hours.        

Let f be the frequency

First calculate (x-x_bar) where x is each number of hours and x_bar is mean. The value of x_bar i.e. $\frac{}{x}$ = 4.201

For example for the Hour = 1 , and mean = 4.201

Then (x-$\frac{}{x}$) = 1 - 4.201 = -3.201

So calculating this for every number of hour we get:

(x-$\frac{}{x}$)

-3.201

-2.201

-1.201

-0.201

0.799

1.799

2.799

3.799

Next calculate (x-$\frac{}{x}$)². Just take the squares of the above column (x-$\frac{}{x}$)

For example the first entry of below calculation is computed by:

 (x-$\frac{}{x}$)² = (-3.201 )² = 10.246401

  (x-$\frac{}{x}$)²

10.246401

4.844401

1.442401

0.040401

0.638401

3.236401

7.834401

14.432401

Next multiply each entry of  (x-$\frac{}{x}$)² with frequency f. For example the first entry below is computed by:

(x-$\frac{}{x}$)² * f = 10.246401  * 16 = 163.942416

(x-$\frac{}{x}$)² * f

163.942416

164.709634

73.562451

1.575639

21.705634

51.782416

70.509609

432.97203

Now the formula to calculate standard deviation is:

S = √∑(x-$\frac{}{x}$)² * f/n

Here

n = ∑f = 229

∑(x-$\frac{}{x}$)² * f  is the sum of all entries of (x-$\frac{}{x}$)² * f

∑(x-$\frac{}{x}$)² * f = 980.759829

S = √∑(x-$\frac{}{x}$)² * f/n

  = √980.759829  /  229

  = √4.2827940131004

  = 2.0694912449924

S = 2.069

b-2) How long is a typical customer parked?

That is the value of mean calculated in part b-1. Hence

Typical Customer Parked for 4.201 hours

c) Find the mean and the standard deviation of the amount charged.

First multiply each value of Amount Charged by the corresponding frequency. Let x represents the number of hours and f represents frequency. Then:

fx

48

204

612

624

714

384

243

870

Now add the above computed products.

∑fx = 48+204+612+624+714+384+243+870  = 3699

Compute Mean:

Now the formula to calculate mean:

Mean = Sum of the value / Number of value

         = ∑fx / ∑f

         = 3699 / 229

Mean = 16.153

Compute Standard Deviation:

Let x be the Amount Charged.        

Let f be the frequency.

First calculate (x-x_bar) where x is each value of Amount charged and x_bar is mean. The value of x_bar i.e. $\frac{}{x}$ = 16.153

For example for the Amount Charged = 3 , and mean = 16.153

Then (x-$\frac{}{x}$) = 3 - 16.153 = -13.153

So calculating this for every number of hour we get:

(x-$\frac{}{x}$)

-13.153

-10.153

-4.153

-0.153

4.847

7.847

10.847

12.847

Next calculate (x-$\frac{}{x}$)². Just take the squares of the above column (x-$\frac{}{x}$)

For example the first entry of below calculation is computed by:

 (x-$\frac{}{x}$)² = (-13.153  )² = 173.001409

  (x-$\frac{}{x}$)²

173.001409

103.083409

17.247409

0.023409

23.493409

61.575409

117.657409

165.045409

Next multiply each entry of  (x-$\frac{}{x}$)² with frequency f. For example the first entry below is computed by:

(x-$\frac{}{x}$)² * f = 173.001409  * 16 =  

   (x-$\frac{}{x}$)² * f

2768.022544

3504.835906

879.617859

0.912951

798.775906

985.206544

1058.916681

4951.36227

∑(x-$\frac{}{x}$)² = 14947.65066

Now the formula to calculate standard deviation is:

S = √∑(x-$\frac{}{x}$)² * f/n

Here

n = ∑f = 229

∑(x-$\frac{}{x}$)² * f  is the sum of all entries of (x-$\frac{}{x}$)² * f

∑(x-$\frac{}{x}$)² * f = 14947.65066

S = √∑(x-$\frac{}{x}$)² * f/ ∑f

  = √65.273583668122

  = 8.0792068712295

S = 8.079