**Data Sufficiency in Statistics**

A sufficient statistic is a statistic that summarizes all of the information in a sample about a chosen parameter. For example, the sample mean, x̄, estimates the population mean, μ. x̄ is a sufficient statistic if it retains all of the information about the population mean that was contained in the original data points.

Let X1, X2, …, Xn be a random sample from a probability distribution with unknown parameter θ. Then, the statistic:

Y=u(X1,X2,…,Xn)

is said to be sufficient for θ if the conditional distribution of X1, X2, …, Xn, given the statistic Y, does not depend on the parameter θ.

**Guidelines to solve questions**

In each of the questions below consists of a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer.

- If the data in statement I alone are sufficient to answer the question, while the data in statement II alone are not sufficient to answer the question
- If the data in statement II alone are sufficient to answer the question, while the data in statement I alone are not sufficient to answer the question
- If the data either in statement I alone or in statement II alone are sufficient to answer the question
- If the data given in both statements I and II together are not sufficient to answer the question and
- If the data in both statements I and II together are necessary to answer the question.

**Example of Data sufficiency **

Question: In which year was Rahul born ?

Statements:

- Rahul at present is 25 years younger to his mother.
- Rahul’s brother, who was born in 1964, is 35 years younger to his mother.

A. | I alone is sufficient while II alone is not sufficient |

B. | II alone is sufficient while I alone is not sufficient |

C. | Either I or II is sufficient |

D. | Neither I nor II is sufficient |

E. | Both I and II are sufficient |

**Answer E**

**Explanation:**

From both I and II, we find that Rahul is (35 – 25) = 10 years older than his brother, who was born in 1964. So, Rahul was born in 1954.

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