Question

The number of arrivals per minute at a bank located in the central business district of a large city was recorded over a period of 200​ minutes, with the results shown in the table below. Complete​ (a) through​ (c) to the right.

Arrivals	Frequency
0	13
1	24
2	48
3	44
4	30
5	20
6	12
7	6
8	3

a. Compute the expected number of arrivals per minute.

μ=

​(Type an integer or decimal rounded to three decimal places as​ needed.)

b. Compute the standard deviation.

σ=

​(Type an integer or decimal rounded to three decimal places as​ needed.)

c. What is the probability that there will be fewer than 2 arrivals in a given​ minute?

P(x<2)=

​(Type an integer or decimal rounded to three decimal places as​ needed.)

Answer 1

Consider,

X = Arrivals

f = frequency

We prepare the following table using above information in the question

0	13	0	3.050	9.3025	120.9325
1	24	24	3.050	4.2025	100.86
2	48	96	3.050	1.1025	52.92
3	44	132	3.050	0.0025	0.11
4	30	120	3.050	0.9025	27.075
5	20	100	3.050	3.8025	76.05
6	12	72	3.050	8.7025	104.43
7	6	42	3.050	15.6025	93.615
8	3	24	3.050	24.5025	73.5075
	200	610			649.5

a. We compute the expected number of arrivals per minute using following formula

The expected number of arrivals per minute =

b. We compute the standard deviation

(Round answer up to 3 decimal places)

The standard deviation =

c. We find the probability that there will be fewer than 2 arrivals in a given​ minute

----------(1)

We convert above X into Z using following formula

--------(2)

Using equation (2) in equation (1) we get

------------(3)

We find the above value using Excel function

=NORMSDIST(Z)

Here, Z = -0.582685904

=NORMSDIST(-0.582685904) then press Enter we get probability is as below

(Round answer up to 3 decimal places)

Using Excel function

The probability that there will be fewer than 2 arrivals in a given​ minute is 0.280

If we write the equation (3) rounded up to 2 decimal places is as below

We find the the above probability using z table of standard normal curve areas

(Round answer up to 3 decimal places)

Using z table of standard normal curve areas

The probability that there will be fewer than 2 arrivals in a given​ minute is 0.281

Summary :-

a. The expected number of arrivals per minute =

b. The standard deviation =

c. Using Excel function

The probability that there will be fewer than 2 arrivals in a given​ minute is 0.280

or

Using z table of standard normal curve areas

The probability that there will be fewer than 2 arrivals in a given​ minute is 0.281