# Sample Size Formula

In the field of statistics, the sample size is defined as the number of observations used to calculate population estimates for a specific population. In other words, it refers to the number of individual samples utilized in a data study. It uses the difference between the population and the sample to calculate the correct sample size. It is the process of selecting a group of people from a population to estimate the characteristics of the entire population, known as sampling. It is denoted by the symbol n.

### Sample Size Formula

For a smaller sample size, the concept of the T distribution is used in place of normal distribution. Specifically, this distribution is used when the value of the sample size is less than 30. In this test, we utilize the t statistic to test the null hypothesis using both one-tailed and two-tailed tests if the population variance is unknown and the sample size is small. It is also known as adjusted sample size.

A = n / (1 + (n – 1)/P)Where,

- A is the adjusted sample size,
- n is the sample size,
- P is the population size.

For infinite population size, the formula is expressed in terms of z-value and error margin.

n = Z^{2}p(1 – p)/m^{2}Where,

- n is the sample size,
- Z is the z-value,
- p is the proportion of population (generally taken as 0.5),
- m is the margin of error.

### Sample Problems

**Problem 1: Calculate the adjusted sample size for a sample size of 300 and a population of 50000.**

**Solution:**

We have,

n = 300

P = 50000

Using the formula we have,

A = n / (1 + (n – 1)/P)

= 300 / (1 + 299/50000)

= 300/1.00598

= 298.216

**Problem 2: Calculate the adjusted sample size for a sample size of 100 and a population of 25000.**

**Solution:**

We have,

n = 100

P = 25000

Using the formula we have,

A = n / (1 + (n – 1)/P)

= 100 / (1 + 299/25000)

= 100/1.001196

= 99.88

**Problem 3: Calculate the adjusted sample size for a sample size of 76 and a population of 2000.**

**Solution:**

We have,

n = 76

P = 2000

Using the formula we have,

A = n / (1 + (n – 1)/P)

= 76 / (1 +75/2000)

= 76/1.0375

= 73.25

**Problem 4: Calculate the population size if the adjusted sample size is 102.2 for a sample size of 104.**

**Solution:**

We have,

A = 102.2

n = 104

Using the formula we have,

A = n / (1 + (n – 1)/P)

=> 102.2 = 104 / (1 + 103/P)

=> 1 + 103/P = 1.01

=> 103/P = 0.01

=> P = 10300

**Problem 5: Calculate the sample size for z-value as 1.5 and the margin of error as 4.2%.**

**Solution:**

We have,

z = 1.5

m = 4.2% = 0.042

p = 0.5

Using the formula we have,

n = Z

^{2}p(1 – p)/m^{2}= (1.5)

^{2}× 0.5 × (1 – 0.5)/(0.042)^{2}= 0.5625/0.001764

= 318.87

**Problem 6: Calculate the sample size for z-value as 1.2 and the margin of error as 3.5%.**

**Solution:**

We have,

z = 1.2

m = 3.5% = 0.035

p = 0.5

Using the formula we have,

n = Z

^{2}p(1 – p)/m^{2}= (1.2)

^{2}× 0.5 × (1 – 0.5)/(0.035)^{2}= 0.36/0.001225

= 293.877

**Problem 7: Calculate the z-value if the sample size is 250 and the margin of error is 3.2%.**

**Solution:**

We have,

n = 250

m = 3.2% = 0.032

p = 0.5

Using the formula we have,

n = Z

^{2}p(1 – p)/m^{2}=> Z

^{2}= nm^{2}/(p(1 – p))=> Z

^{2}= 250 × (0.032)^{2}/ (0.5 × 0.5)=> Z

^{2}= 0.256/0.025=> Z

^{2}= 10.24=> Z = 3.2