Business Statistics Problems

Chapter 73

Problem 7.93

Problem 7.215

Chapter 89

Problem No. 8.99

Problem No. 8.1711

Chapter 1312

Problem No. 13.912

Problem No. 13.2115

Chapter 1616

Problem No. 16.316

Problem No. 16.717

References21

Chapter 7

Problem 7.9

A.

Here the method of stratified sampling is used because the sample is divided into homogenous sub groups. In this type of sampling the population is divided into homogenous subgroups and after that a simple random sample from each group is taken. The sub groups in which the population is divided are called strata.

Mathematically,

Nh:- Total number of population units in stratum h.

nh:- Total number of sample units in stratum h.

N: Total number of population

n: sample size

f=n/N : sampling fraction

The stratified random sampling not only represents the whole population but also represents the sub groups within the population through strata. There are two types of stratified sampling, proportional sampling and disproportional sampling. In proportional stratification the sampling fraction is equal in each strata and in disproportional stratification the sampling fraction in not equal across strata.

B.

The division of samples into different strata is given below.

 

No. of Invoices

Percentage in the total

Stratum 1

50

1.00%

Stratum 2

500

10.00%

Stratum 3

1000

20.00%

Stratum 4

3450

69.00%

Total Invoices

5000

100.00%

Since the sampling fraction in each strata is unknown, the method of disproportional stratifies sampling will be used.

The total no. of invoices is N=5000. The sample size required is n=500. 1% of the total invoices are placed in stratum 1 i.e. n1=50, 10% of the total invoices are placed in stratum 2 i.e. n2=500, 20% of the total invoices are placed in stratum 3 i.e. n3=1000 and 69% of the total invoices are placed in stratum 4 i.e. n4=3450. The units in each sample are then assigned a code respective to the no. of digits they are composed of. The code assigned to n1 is from 01 to 50, the code assigned to n2 is from 001 to 500, the code assigned to n3 is from 0001 to 1000 and the code assigned to n1 is from 0001 to 3450.

The percentage composition for the required stratified sample that is proportional to the strata size will consist of 1% from stratum 1, 10% from stratum 2, 20% from stratum 3 and 69% from stratum 4. This means that four different simple random samples will be taken from the four different strata using random numbers. In the first sample 5 invoices will be selected form the stratum 1, 50 invoices from the stratum 2 for the second sample, 100 invoices from the stratum 3 for the third sample, and 345 invoices from the stratum 4 for the fourth sample. These different samples will then be combined to obtain results that reflect the homogeneity of different groups or strata.

C.

In simple random sampling, a sample from the whole population is taken, but it does not show the exclusivity of subgroups within the population. For this reason stratified random sampling is used. In each strata of the stratified random sampling every nh unit has equal chance of occurring (through simple random sampling) and that is why stratified sampling is random.

Problem 7.21

A.

Here mean= 8 minutes and ...