Content Writer
The sample size formula is used to determine the size of the population through the difference between the total population and the sample. It is denoted by “n” or “N”.
- The sample size formula takes into consideration the number of observations to estimate a specific population.
- It is considered useful because of the difficulty faced in calculating the whole population.
- So, to make the process easier, surveys are conducted.
- We use factors like level of confidence and margin of error to determine whether the sample size is accurate or not.
- The sample size formula is used for two cases, which are infinite (unknown) population and finite (known) population.
- When a survey has already been conducted, it will determine the variations in response using the standard deviation.
- The formula can be used to determine whether a Samsung phone is real or not.
- Mathematically, the sample size formula can be represented as:
N = [Z2p (1 − p)]/ C2
N/ [1 + {(N − 1)/Pop}]
Key Terms: Sample, Sample Size Formula, Infinite, Finite, Population, Confidence, Margin, Survey, Report, Variables, Constraint, Standard deviation
What is Sample Size Formula?
[Click Here for Sample Questions]
Sample Size Formula is a formula that is used to determine the accurate size of the sample by using the difference between the population and the sample.
- Sample Size is an important concept used in the field of statistics.
- It analyses the number of sample used in the data study.
- Confidence level, standard deviation and Z value are the factors used in sample size.
- The process help in understanding target audience and derive better conclusions.
- Population consists of all data required for research purpose.
- The formula is used for finite and infinite population.
Example of What is Sample Size Formula?Example 1: Suppose we conduct a study of 100 people who are interested in eating ice cream of vanilla flavour, then the sample size is 100. Example 2: Determine the adjusted sample size for a sample size of 200 and a population of 40000. Ans: We have,
|
Sample Size Formula
Also Read:
| Related Articles | ||
|---|---|---|
| Polynomials | Polynomials Formulas | Area of Sector |
| Roots of polynomials | Degree of polynomials | Value of e |
Sample Size Formula for Infinite and Finite Population
[Click Here for Sample Questions]
The sample size Formula for infinite and finite population are as follows:
For Infinite population
When the data is collected from infinite population. Then, the sample size formula is
N = [Z2p (1 − p)]/ C2
Where
- N represents the Sample size and p is the percentage of population.
- Moreover, C is confidence level whereas Z value is given.
Example of Infinite PopulationExample: Calculate the sample size for z-value as 0.5 and the margin of error as 3.2%? Ans: Given, z = 0.5
|
For Finite population
When the data collected from the finite or known population. Then, the sample size formula is given by
N/ [1 + {(N − 1)/Pop}]
Where
- N is sample size for infinite population
- Pop represents the population.
- The given figure shows the formula of sample size in the form of standard deviation.
Example of Finite PopulationExample: Determine the adjusted sample size for a sample size of 100 and a population of 10000. Ans: We have,
|
How to apply Sample Size Formula?
[Click Here for Sample Questions]
The process of calculating the whole population, one should follow the given steps accordingly.
Step 1: Take a survey
The foremost step of the process is to conduct a census or survey among the people as per the requirement of the issue. It is advised to survey small populations because it will reduce time, effort and money.
- An individual can also use or take help from similar surveys held by different organizations.
- But the data should be rechecked.
- If the error is carried forward in calculating the sample size, then the desired result is unlikely to be found.
Step 2: Find out other constraint’s values required in the formula
The sample size formula contains constraints like confidence level, Z value or percentage of population. So, to calculate the above constraint, a surveyor should go through the different types of tables.
- Moreover, the percentage of the population can be easily calculated by the use of the percentage formula.
Step 3: Applying the formula
There are two formulas for calculating the sample size based on the type of population. An individual should choose the correct formula to apply as per the survey taken.
- Next, put the values of the variables in the equation.
- Performing basic mathematics will give the value of the required sample size.
Things to Remember
- The sample size formula differs for different population types.
- It involves the selection of a group of people from the total population to calculate the features of the entire population.
- Z score is dependent on the value of confidence level.
- 10% of the total population is considered a good sample size.
- Study design, outcome measures and method of sampling affect sample size.
- The sample size formula is used for finite and infinite populations.
- As the sample size increases, the sampling distribution approaches a normal distribution.
Also Read:
| Related Articles | ||
|---|---|---|
| Surface Area of a Cylinder | Trigonometric Ratios | Algebra |
| Trigonometry | Centroid | Value of cos 60 |
Sample Questions
Ques: Explain sample size with an example? (3 marks)
Ans: In mathematics or other fields, sample size is the number of people observed from whole population for some survey. These people are the representative for the whole.
- For example, let us suppose that new setup mobile company wants to take the feedback from the users.
- So, they take a small proportion and survey them.
- Here, the number of people surveyed are sample size.
Ques: How to calculate sample size? (2 marks)
Ans: It is very tricky to calculate the sample size of the population because of the confidence level and margin of error in the survey conducted. After observing the population, there are four commonly used formula to calculate sample size which are sample size formula, Cochran’s formula, Yamane’s formula and Slovin’s formula.
Ques: Define the following terms:
(a) Marginal error
(b) Confidence level? (3 marks)
Answer: Margin of error is the amount that gives width to the estimated percentage to get accuracy. It provides the range where the value may occur. For example, the margin of error in survey is 2%. So, the observed value 53 can be expressed as 53 ± 2 percent.
- Confidence level is the probability that the proportion that is true is contained by the margin of error.
- The higher the confidence level, the more certain you can be that the interval includes the true ratio.
Ques: How is Z score determined in sample size formula? (3 marks)
Ans: The sample size formula is given by
N = [Z2p (1 − p)]/ C2
Where z is the value calculated based upon the confidence level C. Z-score is a numerical measurement used to describe a value's relationship to the mean of a group of values. Z-score is measured in terms of standard deviations of values from their mean.
Ques: Differentiate Slovin’s Formula from Sample size formula? (2 marks)
Ans: Both Slovin’s formula and Sample size formula are used to determine the sample size of the population but Slovin’s formula is different from the sample size formula because it allow us to sample the population with a desired degree of accuracy. If an individual need more accurate value, then it is advised to use Slovin’s formula as it indicates the size of sample population to be taken in account for better results.
Ques: Find the sample size for some finite and infinite population when the percentage of 3500 population is given as 0.05. Here, take the confidence level as 99 and confidence interval as 0.01? (3 marks)
Ans: We know that, the sample size formula for infinite population is
N = [Z2p (1 − p)]/ C2
Here, p = 0.05, C = 0.01 and Z value is taken from z table which is 2.58.
Applying the formula,
N = [(2.58)2(0.05) (1 – 0.05)]/ (0.01)2
N = 3161.8
Also, the sample size formula for finite population denoted by SS’ is
SS’ = N/ [1 + {(N − 1)/Pop}]
Here, N = 3161.8 and Pop = 3500
Putting value in the formula,
SS’ = 3161.8/ [1 + {(3161.8 − 1)/3500}]
SS’ = 1661.5
Ques: 41% of Jacksonville residents said that they had been in a hurricane. How many adults should be surveyed to estimate the true proportion of adults who have been in a hurricane, with a 95% confidence interval 6% wide? (3 marks)
Ans: We know that, the sample size formula for infinite population is
N = [Z2p (1 − p)]/ C2
Here, p = 41% = 0.41, C = 6%/2 = 0.03 and Z value is taken from z table which is 1.96.
Applying the formula,
N = [(1.96)2(0.41) (1 – 0.41)]/ (0.03)2
N = [(1.96)2(0.41) (0.59)]/ (0.03)2
N = 1032.5
Sample size is 1033.
Ques: Calculate the sample size for a population of 100000. Take confidence level as 95% and margin of error as 5%? (3 marks)
Ans: We know that, the sample size formula for infinite population is
N = [Z2p (1 − p)]/ C2
Here, p = 0.5 C = 0.05 and Z value is taken from z table which is 1.96.
Applying the formula,
N = [(1.96)2(0.5) (1 – 0.5)]/ (0.05)2
N = [(1.96)2(0.5) (0.5)]/ (0.05)2
N = 384.16
Sample size for infinite population is 384.16.
Also, the sample size formula for finite population denoted by SS’ is
SS’ = N/ [1 + {(N − 1)/Pop}]
Here, N = 384.16 and Pop = 100000
Putting value in the formula,
SS’ = 384.16/ [1 + {(384.16 − 1)/100000}]
SS’ = 382.69
Sample size for finite population of 100000 is 383.
Ques: Let us take the example of a retailer who is interested to know how many of their customers bought an item from them after viewing their website on a certain day. Given that their website has on average, 10,000 views per day determine the sample size of the customers that they have to monitor at a 95% confidence level with a 5% margin of error if:
(a) They are uncertain of the current conversion rate.
(b) They know from previous surveys that the conversion rate is 5%? (5 marks)
Ans: We know that, the sample size formula for infinite population is
N = [Z2p (1 − p)]/ C2
Here, C = 0.05 and Z value is taken from z table which is 1.96.
Also, the current conversion rate is unknown so let p = 0.5
Applying the formula,
N = [(1.96)2(0.5) (1 – 0.5)]/ (0.05)2
N = [(1.96)2(0.5) (0.5)]/ (0.05)2
N = 384.16
Sample size for infinite population is 384.16.
Also, the sample size formula for finite population denoted by SS’ is
SS’ = N/ [1 + {(N − 1)/Pop}]
Here, N = 384.16 and Pop = 10000
Putting value in the formula,
SS’ = 384.16/ [1 + {(384.16 − 1)/10000}]
SS’ = 370 (approx.)
Sample size for finite population of 10000 is 370.
- We know that, the sample size formula for infinite population is
N = [Z2p (1 − p)]/ C2
Here, p = 0.05, C = 0.05 and Z value is taken from z table which is 1.96.
Applying the formula,
N = [(1.96)2(0.05) (1 – 0.05)]/ (0.05)2
N = [(1.96)2(0.05) (0.95)]/ (0.05)2
N = 72.99 or 73
Sample size for infinite population is 72.99 or 73.
Also, the sample size formula for finite population denoted by SS’ is
SS’ = N/ [1 + {(N − 1)/Pop}]
Here, N = 73 and Pop = 10000
Putting value in the formula,
SS’ = 73/ [1 + {(73 − 1)/10000}]
SS’ = 72.47 (approx.)
Sample size for finite population of 10000 is 72.
Ques: Find the sample size for some finite and infinite population when the percentage of 4300 population is given as 0.05. Here take confidence level as 99 and confidence interval as 0.01? (4 marks)
Ans: We know that, the sample size formula for infinite population is
N = [Z2p (1 − p)]/ C2
Here, p = 0.05 C = 0.01 and Z value is taken from z table which is 2.58.
Applying the formula,
N = [(2.58)2(0.05) (1 – 0.05)]/ (0.01)2
N = [(2.58)2(0.05) (0.95)]/ (0.01)2
N = 3161.8
Sample size for infinite population is 3161.8.
Also, the sample size formula for finite population denoted by SS’ is
SS’ = N/ [1 + {(N − 1)/Pop}]
Here, N = 3161.8 and Pop = 4300
Putting value in the formula,
SS’ = 3181.6/ [1 + {(3181.8 − 1)/4300}]
SS’ = 852
Sample size for finite population of 4300 is 852.
Ques: Suppose we want to know the average age of a Florida State College student, plus or minus 0.5 years. We’d like to be 99% confident about our result. Here, we know that the standard deviation for the population is 2.9? (5 marks)
Ans: Step 1: find the value of Z score
Here, confidence level C = 99% = 0.99
Divide the confidence level by 2 which gives the value 0.495. Now, use the z score table to find closest value of z to 0.495 which is 2.58.
Hence, Z = 2.58
Step 2: Multiply the standard deviation with the Z.
We know, the standard deviation for the population is 2.9 and Z score is 2.58.
Standard deviation × Z-score = 2.9 × 2.58
Standard deviation × Z score = 7.482
Step 3: Divide the value of resultant multiple value of standard deviation and Z-score with margin of error.
Here, Standard deviation × Z score = 7.482 and margin of error = 0.5
After dividing, the value is 14.96.
Step 4: Square the value of the resultant from step 3.
Therefore 14.96 × 14.96 = 223.8016
Hence, the sample size is 223.8016.
Ques: Calculate the adjusted sample size for a sample size of 200 and a population of 15000? (3 marks)
Ans: Given, n = 200
- P = 15000
- Using the formula we have,
- A = n / (1 + (n – 1)/P)
- 200 / (1 + 199/15000)
- 200/1.013
- 197.43
Ques: Calculate the adjusted sample size for a sample size of 10 and a population of 1500? (3 marks)
Ans: Given,
- n = 10
- P = 1500
- Using the formula we have,
- A = n / (1 + (n – 1)/P)
- 10 / (1 + 9/1500)
- 10/1.006
- 9.940
Ques: Calculate the adjusted sample size for a sample size of 70 and a population of 1000? (3 marks)
Ans: Given,
- n = 70
- P = 1000
- Using the formula we have,
- A = n / (1 + (n – 1)/P)
- 70/ (1 +69/1000)
- 70/1.069
- 65.48
Ques: Calculate the population size if the adjusted sample size is 102.2 for a sample size of 204? (3 marks)
Ans: Given,
- A = 102.2
- n = 204
- Using the formula we have,
- A = n / (1 + (n – 1)/P)
- 102.2 = 204 / (1 + 203/P)
- 1 + 203/P = 0.50
- 203/P = 0.50
- P = 406
For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates
Check-Out:
Comments