Respuesta :
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. [tex]\frac{}{x}[/tex] = 4.201
For example for the Hour = 1 , and mean = 4.201
Then (x-[tex]\frac{}{x}[/tex]) = 1 - 4.201 = -3.201
So calculating this for every number of hour we get:
(x-[tex]\frac{}{x}[/tex])
-3.201
-2.201
-1.201
-0.201
0.799
1.799
2.799
3.799
Next calculate (x-[tex]\frac{}{x}[/tex])². Just take the squares of the above column (x-[tex]\frac{}{x}[/tex])
For example the first entry of below calculation is computed by:
(x-[tex]\frac{}{x}[/tex])² = (-3.201 )² = 10.246401
(x-[tex]\frac{}{x}[/tex])²
10.246401
4.844401
1.442401
0.040401
0.638401
3.236401
7.834401
14.432401
Next multiply each entry of (x-[tex]\frac{}{x}[/tex])² with frequency f. For example the first entry below is computed by:
(x-[tex]\frac{}{x}[/tex])² * f = 10.246401 * 16 = 163.942416
(x-[tex]\frac{}{x}[/tex])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-[tex]\frac{}{x}[/tex])² * f/n
Here
n = ∑f = 229
∑(x-[tex]\frac{}{x}[/tex])² * f is the sum of all entries of (x-[tex]\frac{}{x}[/tex])² * f
∑(x-[tex]\frac{}{x}[/tex])² * f = 980.759829
S = √∑(x-[tex]\frac{}{x}[/tex])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. [tex]\frac{}{x}[/tex] = 16.153
For example for the Amount Charged = 3 , and mean = 16.153
Then (x-[tex]\frac{}{x}[/tex]) = 3 - 16.153 = -13.153
So calculating this for every number of hour we get:
(x-[tex]\frac{}{x}[/tex])
-13.153
-10.153
-4.153
-0.153
4.847
7.847
10.847
12.847
Next calculate (x-[tex]\frac{}{x}[/tex])². Just take the squares of the above column (x-[tex]\frac{}{x}[/tex])
For example the first entry of below calculation is computed by:
(x-[tex]\frac{}{x}[/tex])² = (-13.153 )² = 173.001409
(x-[tex]\frac{}{x}[/tex])²
173.001409
103.083409
17.247409
0.023409
23.493409
61.575409
117.657409
165.045409
Next multiply each entry of (x-[tex]\frac{}{x}[/tex])² with frequency f. For example the first entry below is computed by:
(x-[tex]\frac{}{x}[/tex])² * f = 173.001409 * 16 =
(x-[tex]\frac{}{x}[/tex])² * f
2768.022544
3504.835906
879.617859
0.912951
798.775906
985.206544
1058.916681
4951.36227
∑(x-[tex]\frac{}{x}[/tex])² = 14947.65066
Now the formula to calculate standard deviation is:
S = √∑(x-[tex]\frac{}{x}[/tex])² * f/n
Here
n = ∑f = 229
∑(x-[tex]\frac{}{x}[/tex])² * f is the sum of all entries of (x-[tex]\frac{}{x}[/tex])² * f
∑(x-[tex]\frac{}{x}[/tex])² * f = 14947.65066
S = √∑(x-[tex]\frac{}{x}[/tex])² * f/ ∑f
= √65.273583668122
= 8.0792068712295
S = 8.079
