**Which of the following statements is correct?**

** ****Options:**

a) Lower the sampling risk greater the sample size b) Smaller the tolerable error, greater the sample size c) Lower the expected error, smaller the sample size d) All are correct |

### The Correct Answer Is:

d) All are correct

**The correct answer is indeed option “d) All are correct.”**

Let’s delve into the details to understand why this answer is correct and why the other options are not:

**a) Lower the sampling risk, greater the sample size:**

This statement is correct. Sampling risk refers to the risk that the sample you select may not accurately represent the entire population. By increasing the sample size, you can reduce this risk.

A larger sample size provides more data points and is more likely to reflect the true characteristics of the population, which helps in minimizing sampling errors and risk.

**b) Smaller the tolerable error, greater the sample size:**

This statement is also correct. Tolerable error, also known as margin of error, is the maximum acceptable difference between the sample estimate and the true population parameter. If you want a smaller margin of error, you’ll need a larger sample size.

This is because a larger sample size provides more data points and, as a result, allows for more precise estimation, reducing the margin of error.

**c) Lower the expected error, smaller the sample size:**

This statement is correct as well. The expected error, in the context of statistical sampling, is the anticipated difference between the sample estimate and the true population parameter. If you aim for a lower expected error, you can achieve it with a smaller sample size if the population is homogenous and variability is low.

In cases where the population is not very diverse, a smaller sample can provide accurate estimates. However, in cases with high variability or diverse populations, a larger sample size is needed to reduce expected error.

**Now, let’s explain why the other options are not correct:**

**a) “Lower the sampling risk, greater the sample size”:**

This statement is correct, as explained above. Increasing the sample size reduces sampling risk. However, it’s important to note that this relationship is not linear. After a certain point, increasing the sample size may have diminishing returns in terms of reducing sampling risk.

This is because there is a point of diminishing returns where additional data points do not significantly improve the accuracy of the estimate.

**b) “Smaller the tolerable error, greater the sample size”:**

This statement is also correct. When you want a smaller margin of error, you need a larger sample size to achieve more precise estimates. This is because a smaller margin of error requires more data points to make sure that the estimate is within a very narrow range of the true population parameter.

**c) “Lower the expected error, smaller the sample size”:**

This statement is correct only under specific conditions. If the population is highly homogenous and has low variability, a smaller sample size can lead to lower expected error. In such cases, a smaller sample size may be sufficient to provide a reliable estimate.

However, in situations where the population is diverse or exhibits high variability, a smaller sample size is likely to lead to a higher expected error. This is because a smaller sample may not adequately capture the range of variation present in the population.

It’s important to understand that these relationships are not absolute and can be influenced by various factors, including the nature of the population, the research question, and the level of precision required.

Therefore, while these statements provide general guidelines, they should be applied with careful consideration of the specific context and characteristics of the population being studied.

In summary, all of the given statements are correct when applied in the appropriate context of statistical sampling and estimation. However, it’s crucial to recognize that these relationships are not universal and may be subject to specific conditions and considerations.

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