MODULE 4

MODULE 4

Random Sampling, Parameter and Statistic, and Sampling Distribution of Statistics

Icebreaker

This icebreaker helps you understand why sampling is necessary. 

Questions:

1. How many penguins are in the picture.

2. How long will it take them to come up with a number.

3. Can you suggest a better way to count the penguins.

By placing a grid over the image helps with systematic sampling to quickly obtain an accurate estimate. 

Pretest

Choose the correct answer

Multiple Choice

It refers to the entire group that is under study or investigation

• Population

• Sample

• Random Sampling

• Subset

Multiple Choice

It is a subset taken from a population, either by random or non-random

• Population

• Sample

• Random Sampling

• Lottery Sampling

Multiple Choice

What sampling technique in which members of the population are listed and samples are selected in intervals call sample intervals.

• Systematic Sampling

• Lottery Sampling

• Stratified Random Sampling

• Quota Sampling

Multiple Choice

It is sometimes called area sampling, it applies on a geographical basis.

• Systematic Sampling

• Lottery Sampling

• Stratified Random Sampling

• Cluster Sampling

Multiple Choice

It refers to a part of the sampling technique where each sample point has an equal chance of being selected.

• Systematic Sampling

• Lottery Sampling

• Random Sampling

• Quota Sampling

What is Sampling?

Sampling allows us to estimate the parameters of a population by studying only a small selection or sample of the population. When the population is not uniform, the number of samples and the method of choosing samples are critical.

What is Sampling?

• Often it is impractical to examine an entire population.

• Instead, we select a sample from our population of interest and, on the basis of this sample, information about the entire population will be inferred.

Notes

Variation between the individual in a population leads to difficulties if attempts are made to generalize results based on a sample. If it were not for this ever-present variation a sample consisting of a single value would provide all the information about the population. An extensive theory of sampling techniques has been developed to assist in establishing meaningful inferences about a total population from the values contained in a sample.

Random Sampling

Random sampling is a part of the sampling technique in which each sample has an equal probability of being chosen.

Types of Random Sampling

There are four types of random sampling: Simple Random Sampling, Systematic Sampling, Stratified Sampling, and Cluster Sampling

Simple Random Sampling

Simple random sampling is a type of probability sampling in which the researcher randomly selects a subset of participants from a population. Each member of the population has an equal chance of being selected. Data is then collected from as large a percentage as possible of this random subset.

Systematic Sampling

Systematic sampling is a type of probability sampling method in which sample members from a larger population are selected according to a random starting point but with a fixed, periodic interval. This interval, called the sampling interval, is calculated by dividing the population size by the desired sample size.

Stratified Sampling

Stratified random sampling is a method of sampling that involves the division of a population into smaller sub-groups known as strata. In stratified random sampling or stratification, the strata are formed based on members' shared attributes or characteristics such as income or educational attainment.

Stratified Sampling

Stratified random sampling is also called proportional random sampling or quota random sampling.

Cluster Sampling

Cluster sampling is a sampling plan used when mutually homogeneous yet internally heterogeneous groupings are evident in a statistical population. It is often used in marketing research. In this sampling plan, the total population is divided into these groups and a simple random sample of the groups is selected.

Advantages and Disadvantages

Every sampling method has pros and cons

Advantages of Simple Random Sampling

• Representativeness and freedom from bias

• Ease of sampling and analysis

Disadvantages of Simple Random Sampling

• Errors in sampling

• Time and labor requirements

Advantages of Systematic Sampling

• Spatially well distributed

• Small standard errors

• Long history of use

Disadvantages of Systematic Sampling

• Bias in overestimating the actual standard error

• Less flexible to increase or decrease the sampling size

• Not applicable for fragmented strata

Advantages of Stratified Sampling

• Allows specifying the sample size within each stratum

• Allows for different sampling design for each stratum

Disadvantages of Stratified Sampling

• Yields large standard error if the sample size selected is not appropriate

• Not effective if all variables are equally important

Advantages of Cluster Sampling

• Can reduce the time and expense of sampling by reducing travel distance

Disadvantages of Cluster Sampling

• Can yield higher sampling error

• Can be difficult to select representative clusters

That's all for Random Sampling

Up Next: Parameter and Statistic

Pretest

Determine the statement whether it is true or false.

Multiple Choice

A statistic is a number that describes a sample.

• True

• False

Multiple Choice

A parameter is a descriptive measure of population.

• True

• False

Multiple Choice

An example of a parameter is the sample mean.

• True

• False

Multiple Choice

The value of a parameter can be approximated and is not necessarily equal to the statistic of a sample.

• True

• False

Multiple Choice

An example of a statistic is a population mean.

• True

• False

What is Parameter

A parameter is a descriptive population measure. It is a measure of the characteristics of the entire population (a mass of all the units under consideration that share common characteristics) based on all the elements within that population.

Examples of the Parameter

1. All people living in one city, all-male teenagers worldwide, all elements in a shopping cart, and all students in a classroom.

2. The researcher interviewed all the students of a school for their favorite apparel brand.

What is Statistic

Statistic is the number that describes the sample. It can be calculated and observed directly. The statistic is a characteristic of a population or sample group. You will get the sample statistic when you collect the sample and calculate the standard deviation and the mean. You can use sample statistic to draw certain conclusions about the entire population.

Examples of the Statistic

1. Fifty percent of people living in the U.S. agree with the latest health care proposal. Researchers can’t ask hundreds of millions of people if they agree, so they take samples or part of the population and calculate the rest.

2. Researcher interviewed the 70% of COVID-19 survivors.

Conclusion

• A statistic is a descriptive measure computed from a sample data.

• A parameter is a descriptive measure computed from an entire population of data.

That's all for Parameter and Statistic

Up Next: Sampling Distribution of Statistics

Pretest

Given the set of numbers, compute for the mean. Write your answer on the space provided in each item. Round off your answers to two decimal places.

Open Ended

Number 1 -- 4, 12, 34, 45, 6

Open Ended

Number 2 -- 23, 45, 67, 89, 21, 11

Open Ended

Number 3 -- 88, 87, 86, 89, 88, 90

Open Ended

Number 4 -- 34, 21, 45, 67, 23

Open Ended

Number 5 -- 12, 9, 6, 5, 32, 40

Answers

1. 20.2

2. 42.67

3. 88

4. 38

5. 71.33

What is Sampling Distribution

The sampling distribution of a statistic is a probability distribution based on a large number of samples of size n from a given population.

Example #1

A population consists of five numbers 2, 3, 6, 10, and 12. Consider samples of size 2 that can be drawn from this population.

How many possible samples can be drawn?

To answer this, use the formula _NC_n (the number of N objects taken n at a time), where N is the total population and n is the sample to be taken out of the population,

Solution #1

In this case N= 5 and n= 2

₅C₂ = 10

So, there are 10 possible samples to be drawn.

Construct the sampling distribution of sample means.

List all the possible outcome and get the mean of every sample.

Note

Observe that the means vary from sample to sample. Thus, any mean based on the sample drawn from a population is expected to assume different values for samples.

Solution #2

This time, let us make a probability distribution of the sample means. This probability distribution is called the sampling distribution of the sample means.

Answer

Observe that all sample means appeared only one; thus, their probability is P(x)=1/10 or 0.1.

Note

A sampling distribution of sample mean is a frequency distribution using the means computed from all possible random samples of a specific size taken from a population.

Example #2

Construct a sampling distribution of the sample mean for the set of data below.

86 88 90 95 98

Consider a sample size of 3 that can be drawn from a population.

How many possible samples can be drawn?

To answer this, use the formula _NC_n, where N is the total population and n is the sample to be taken out of the population,

Solution #1

In this case N= 5 and n= 3

₅C₃ = 10

So, there are 10 possible samples to be drawn.

Construct the sampling distribution of sample means.

List all the possible outcome and get the mean of every sample.

Solution #2

This time, let us make a probability distribution of the sample means. This probability distribution is called, the sampling distribution of the sample means.

Answer

Observe that 88, 92, and 93 appeared only once; thus their probability is P(x)= 1/10 or 0.1. Since 90 and 94 appeared twice, their probability is P(x)=2/10 or 0.2. While 91 appeared thrice, their probability is P(x)= 3/10 or 0.3.

This concludes our lesson from Group 4

Thank you for listening attentively to our lesson. The link for the post-test will be posted in in-chat messages and/or group conversations.