Sampling Techniques and Statistical Inference Exercise 8.1

Dear Class 12 Samacheer Kalvi students, here are the Text Book Solutions for Exercise 8.1 Business Maths Chapter 8 Sampling Techniques and Statistical Inference for your reference and study.

You can find links to the other exercises below.

Sampling Techniques and Statistical Inference Important Formulas

Solutions for Sampling Techniques and Statistical Inference Exercise 8.2

Solutions for Sampling Techniques and Statistical Inference Exercise 8.3

Sampling Techniques and Statistical Inference Exercise 8.1

$\text{1. What is population?} \\$

$\text{The group of individuals considered under study is called as population. It}\\$
$\text{refers not only to people but to all items that have been chosen for the study.} \\$

$\text{2. What is sample?} \\$

$\text{A selection of a group of individuals from a population in such a way}\\$
$\text{that it represents the population is called as sample.}\\$

$\text{3. What is statistic?} \\$
$\text{Any statistical measure computed from sample is known as statistic.} \\$

$\text{4. Define parameter.} \\$
$\text{The statistical constants of the population like mean (m), variance}(\sigma^{2})\;//[/latex> [latex] \text{ are referred as population parameters.}\\$

$\text{5. What is sampling distribution of a statistic?} \\$
$\text{Sampling distribution of a statistic is the frequency distribution} \\$
$\text{ which is formed with various values of a statistic computed from} [latex] \text{different samples of the same size drawn from the same population.}\\$

$\text{6. What is standard error?} \\$
$\text{The standard deviation of the sampling distribution of a statistic is known as its Standard Error} \\$

$\text{7. Explain in detail about simple random sampling with a suitable example.} \\$
$\text{In this technique the samples are selected in such a way that each }\\$
$\text{and every unit in the population has an equal and independent}\\$
$\text{chance of being selected as a sample. Simple random sampling may}\\$
$\text{be done, with or without replacement of the samples selected. In a} \\$
$\text{simple random sampling with replacement there is a possibility of}\\$
$\text{selecting the same sample any number of times. So, simple random sampling}\\$
$\text{So in simple random sampling from a population of N [latex] units, the }\\$
$\text{probability of drawing any unit at the first draw is [latex] }\;\frac{1}{N},\\$
$the probability of drawing any unit in the second draw from\\$
$among the available (N −1) units is \frac{1}{(N −1)}and so on. \\$
$\text{without replacement is followed. Two methods commonly used include the}\\$
$\text{lottery method and the Table of random number method.}\\$
$\text{(A) Lottery method}\\$
$\text{This is the most popular and simplest method when the population is finite. }\\$
$\text{container and shuffled thoroughly. Then the required numbers of slips are selected}\\$
$\text{In this method, all the items of the population are numbered on separate slips} \\$
$\text{of paper of same size, shape and colour. They are folded and placed in a } \\$
$\text{for the desired sample size. The selection of items thus depends on chance.}\\$
$\text{For example, if we want to select 10 students, out of 100 students, then we must}\\$
$\text{write the names/roll number of all the 100 students on slips of the same size}\\$
$\text{and mix them, then we make a blindfold selection of 10 students. This method }\\$
$\text{is called unrestricted random sampling, because units are selected from the}\\$
$\text{population without any restriction. This method is mostly used in lottery draws.}\\$
$\text{If the population or universe is infinite, this method is inapplicable.}\\$
$\text{(B) Table of Random number}\\$
$\text{When the population size is large, it is difficult to number all the items }\\$
$\text{in separate slips of paper of same size, shape and colour. The alternative}\\$
$\text{method is that of using the table of random numbers. The most practical,}\\$
$\text{easy and inexpensive method of selecting a random sample can be done through}\\$
$\text{``Random Number Table''. The random number table has been so constructed that} \\$
$\text{ each of the digits 0,1,2,…,9 will appear approximately with the same frequency} \\$
$\text{and independently of each other. Some commonly used tables are:}\\$
$\text{ a. L.H.C. Tippett random number series }\\$
$\text{b. Fisher and Yates random number series}\\$
$\text{c. Kendall and Smith random number series}\\$
$\text{ d. Rand Corporation random number series.}\\$

$\text{8. Explain the stratified random sampling with a suitable example.} \\$
$\text{When the population is heterogeneous or different segments or groups with }\\$
$\text{respect to the variable or characteristic under study, then Stratified Random} \\$
$\text{ Sampling method is studied. First, the population is divided into homogeneous}\\$
$\text{ number of sub-groups or strata before the sample is drawn. A sample is drawn}\\$
$\text{from each stratum at random. Following steps are involved for selecting a random} \\$
$\text{ sample in a stratified random sampling method. } \\$
$\text{(a) The population is divided into different classes so that each stratum will}\\$
$\text{consist of more or less homogeneous elements. The strata are so designed that}\\$
$\text{they do not overlap each other.}\\$
$\text{(b) After the population is stratified, a sample of a specified size is drawn} \\$
$\text{at random from each stratum using Lottery Method or Table of Random Number Method.}\\$
$\text{Stratified random sampling is applied in the field of different legislative }\\$
$\text{areas as strata in election polling, division of districts in a state etc…} \\$
$\text{Example: Suppose you're a principal who wants to survey 100 students about online }\\$
$\text{ learning satisfaction. Your school has 1,000 students divided into these grades:} \\$
$\text{Grade 9: 300 students}\\$
$\text{Grade 10: 250 students}\\$
$\text{Grade 11: 200 students}\\$
$\text{Grade 12: 250 students}\\$
$\text{Step 1: Create strata. The strata are the different grades}\\$
$\text{Step 2: Decide how many to sample from each stratum}\\$

*** QuickLaTeX cannot compile formula:
 \text{Since Grade 9 has 30% of the total population (300 out of 1,000), }\\

*** Error message:
File ended while scanning use of \text@.
Emergency stop.

*** QuickLaTeX cannot compile formula:
 \text{you take 30% of 100 = 30 students from Grade 9. Similarly:} \\

*** Error message:
File ended while scanning use of \text@.
Emergency stop.

$\text{Grade 9 → 30 students}\\$
$\text{Grade 10 → 25 students}\\$
$\text{Grade 11 → 20 students}\\$
$\text{Grade 12 → 25 students}\\$
$\text{Step 3:You randomly select within each group from each grade.} \\$
$\text{(e.g., using a random number generator or drawing names).} \\$
$\text{You now have a stratified random sample that ensures each grade is}\\$
$\text{proportionally represented, reducing sampling bias.} \\$

$\text{9. Explain in detail about systematic random sampling with example.} \\$
$\text{Systematic sampling is a commonly used technique, if the complete and up-to-date}\\$
$\text{list of the sampling units is available. We can arrange the items in numerical,} \\$
$\text{alphabetical, geographical or in any other order. The procedure of selecting} \\$
$\text{the samples starts with selecting the first sample at random, the rest being} \\$
$\text{automatically selected according to some pre-determined pattern. A systematic}\\$
$\text{sample is formed by selecting every item from the population, where k refers}\\$
$\text{to the sample interval. The sampling interval can be determined by dividing} \\$
$\text{the size of the population by the size of the sample to be chosen. That is k}\\$
$= \frac{N}{n},\text{where k is an integer. k =Sampling interval, N = Size of the}\\$
$\text{population, n = Sample size.} \\$
$\text{Procedure for selection of samples by systematic sampling method}\\$
$\text{(i) If we want to select a sample of 10 students from a class of} \\$
$\text{100 students, the sampling interval is calculated as k}\\$
$=\frac{N}{n}\; =\frac{100}{10}=10\\$
$\text{The sampling interval 10 denotes that for every 10 samples one} \\$
$\text{sample has to be selected.} \\$
$\text{(ii) The first sample is selected from the first 10 samples through}\\$
$\text{random selection procedures. }\\$
$\text{(iii) If the selected first random sample is 5, then the rest of the }\\$
$\text{samples are automatically selected by incrementing the value of the }\\$
$\text{sampling interval (k = 10) i.e., 5, 15 , 25, 35, 45, 55, 65, 75, 85, 95.} \\$
$\text{Example: Suppose we have to select 20 items out of 6,000. The procedure}\\$
$\text{is to number all the 6,000 items from 1 to 6,000. The sampling interval}\\$
$\text{is calculated as k} =\frac{N}{n} =\frac{6000}{20} = 300 . \text{The sampling }\\$
$\text{interval 300 denotes that for every 300 samples one sample has to be selected.} \\$
$\text{The first sample is selected from the first 300 samples through random}\\$
$\text{selection procedures. If the selected first random sample is 50, then the}\\$
$\text{rest of the samples are automatically selected by incrementing the value of}\\$
$\text{the sampling interval (k=300) ie,50, 350, 650, 950, 1250, 1550, 1850, 2150, }\\$
$\text{2450, 2750, 3050, 3350, 3650, 3950, 4250, 4550, 4850, 5150, 5450, 5750. Items}\\$
$\text{bearing those numbers will be selected as samples from the population.} \\$
$\text{10. Explain in detail about sampling error.} \\$
$\text{Errors, which arise in the normal course of investigation or enumeration on } [latex] \text{account of chance, are called sampling errors. Sampling errors are inherent } [latex] \text{in the method of sampling. They may arise accidentally without any bias or }\\$
$\text{prejudice. Sampling Errors arise primarily due to the following reasons:} \\$
$\text{(a) Faulty selection of the sample instead of correct sample by defective} \\$
$\text{sampling technique.} \\$
$\text{(b) The investigator substitutes a convenient sample if the original sample}\\$
$\text{is not available while investigation.} \\$
$\text{(c) In area surveys, while dealing with border lines it depends upon the }\\$
$\text{investigator whether to include them in the sample or not. This is known as}\\$
$\text{ Faulty demarcation of sampling units.} \\$

$\text{11. Explain in detail about non-sampling error.} \\$
$\text{The errors that arise due to human factors which always vary from one investigator }\\$
$\text{to another in selecting, estimating or using measuring instruments( tape, scale)} \\$
$\text{are called Non-Sampling errors.It may arise in the following ways:} \\$
$\text{(a) Due to negligence and carelessness of the part of either investigator or respondents.} \\$
$\text{(b) Due to lack of trained and qualified investigators.} \\$
$\text{(c) Due to framing of a wrong questionnaire.} \\$
$\text{(d) Due to apply wrong statistical measure}\\$
$\text{(e) Due to incomplete investigation and sample survey.} \\$

$\text{12. State any two merits of simple random sampling.} \\$
$\text{1. Personal bias is completely eliminated.}\\$
$\text{2. This method is economical as it saves time, money and labour.}\\$

$\text{13. State any three merits of stratified random sampling.} \\$
$\text{(1) A stratified random sample can be kept small in size without losing its accuracy.} \\$
$\text{(2) It is easy to administer,if the population under study is sub-divided.} \\$
$\text{(3) It reduces the time and expenses in dividing the strata into geographical divisions,} \\$
$\text{since the government itself had divided the geographical areas.} \\$

$\text{14. State any two demerits of systematic random sampling.} \\$
$\text{1. Systematic samples are not random samples.}\\$
$\text{2. If N is not a multiple of n, then the sampling interval (k) cannot be }\\$
$\text{an integer, thus sample selection becomes difficult.}\\$

$\text{15. State any two merits for systematic random sampling.} \\$
$\text{1. This method distributes the sample more evenly}\\$
$\text{ over the entire listed population.}\\$
$\text{2. The time and work is reduced much.} \\$

$\text{16. Using the following Tippet's random number table.} \\$

2952	6641	3992	9792	7969	5911	3170	5624
4167	9524	1545	1396	7203	5356	1300	2693
2670	7483	3408	2762	3563	1089	6913	7991
0560	5246	1112	6107	6008	8125	4233	8776
2754	9143	1405	9025	7002	6111	8816	6446

$\text{Draw a sample of 10 three digit numbers which are even numbers.} \\$

2952	6641	3992	9792	7969	5911	3170	5624
4167	9524	1545	1396	7203	5356	1300	2693
2670	7483	3408	2762	3563	1089	6913	7991
0560	5246	1112	6107	6008	8125	4233	8776
2754	9143	1405	9025	7002	6111	8816	6446

$\text{Tippet's random numbers are 4 digit numbers. So we can select the first 3 digits }\\$
$\text{which are even numbers from the 4 digit random number. The required random}\\$
$\text{samples are bold formatted and highlighted in orange.}\\$

$\text{17. A wholesaler in apples claims that only} \; 4\% \; \text{of the apples supplied by him} \\$
$\text{ are defective. A random sample of 600 apples contained 36 defective apples. } \\$
$\text{Calculate the standard error concerning of good apples.}\\$

$n=600\\$
$p=4\%=0.04\\$
$q=1-0.04=0.96 \; since\; p+q=1\\$
$S.E (p)=\sqrt{\frac{pq}{n}}\\$
$=\sqrt{\frac{0.04\times0.96}{600}}\\$
$S.E (p)=0.008\\$

$\text{18. A sample of 1000 students whose mean weight is 119 lbs(pounds) from a school}\\$
$\text{in Tamil Nadu State was taken and their average weight was found to be 120 lbs with}\\$
$\text{a standard deviation of 30 lbs. Calculate standard error of mean.}\\$

$n=1000\\$
$\sigma=30\\$
$S.E(\bar{x})=\frac{\sigma}{\sqrt{n}}\\$
$=\frac{30}{\sqrt{1000}}\\$
$S.E(\bar{x})=0.9487\\$

$\text{19. A random sample of 60 observations was drawn from a large population and its}\\$
$\text{standard deviation was found to be 2.5. Calculate the suitable standard error that}\\$
$\text{this sample is taken from a population with standard deviation 3?}\\$

$n=60\\$
$\sigma=3\\$
$S.E(S)=\sqrt{\frac{\sigma^{2}}{2n}}\\$
$=\sqrt{\frac{3^{2}}{2\times60}}\\$
$S.E(S)=0.2739\\$

$\text{20. In a sample of 400 population from a village 230 are found to be eaters of}\\$
$\text{vegetarian items and the rest non-vegetarian items. Compute the standard error } \\$
$\text{assuming that both vegetarian and non-vegetarian foods are equally popular in that village?} \\$

$n=400\\$
$p=\frac{230}{400}\\$
$p=0.58\\$
$q=0.42\; since \; p+q=1\\$
$S.E (p)=\sqrt{\frac{pq}{n}}\\$
$=\sqrt{\frac{0.58\times0.42}{400}}\\$
$=0.246779\\$
$S.E (p)=0.25\\$

Sampling Techniques and Statistical Inference Exercise 8.1

Sampling Techniques and Statistical Inference Exercise 8.1

Related Posts:

Leave a Reply Cancel reply

Recent

Sampling Techniques and Statistical Inference Exercise 8.1

Related Posts:

Reader Interactions

Leave a Reply Cancel reply

Footer

Recent

Tags