# Statistical 'Inference'

• buddingscientist
In summary, the conversation covers various topics in statistics, including the normal distribution, independent and identically distributed random variables, chi-squared distribution, and consistency. The speakers discuss how to approach certain problems and share tips for solving them, such as using the standard deviation to find the distribution of a random variable and utilizing the theorem for transforming a chi-squared distribution into a Beta distribution.
buddingscientist
1. If X1, X2, X3, X4 are a random samle from a normal distribution with mean 17, then what is the distribution of $2(X - 10) / S$ where X should be X bar.

Our notes are just awful for this topic.
Any tips how to proceed with this one, and what is S?

2. Let X1, X2, ... be a sequence of independant and identically distributed random variables with mean zero and variance 36. What is the sequece Cn for which

lim p -> inf $P ( (X / Cn) \leq x) = P (Z \leq x), Z ~ N(0,1) ?$

Would this just be sqrt(Var) = sqrt(36) = 6 ?

4. What is the distribution of X/Y where X and Y are independant ${X^{2}}_{1}$ random variables? (chi-squared with 1 deg. of freedom)

Not sure how to progress with this one, do we play with the degrees of freedom to get some sort of trivial answer?

5. Is theta-hat consistent if MSE(theta-hat) = $e^{1/n} -1$.

Since MSE approaches 0 as n approaches infinity (n-> inf, MSE -> $e^{0} -1$ -> 0) then yes, theta-hat is consistent. ?

1. S is most likely to be standard deviation
2. I have no idea what this question is about
3.
There is a theorem which states that
"if X1 and X2 are two indepdendent chi-squared variates with n1 and n2 d.f resp., then X1/X2 is a Beta2(n1/2,n2/2)"
Direct use of this theorem makes this one easy. I am not sure if u are aware of this theorem. If u want i can prove this one.

-- AI

I think you need to know the standard deviation of X before you can answer 1. S is probably the sample standard deviation, defined as the sum over all i of (Xi - Xbar)/(n - 1).

Your notation on 2 is confusing. I believe you meant: lim p --> inf (P(Xp/Cp < x) = P(Z < x)) Assuming you meant that, it's probably intended that X is distributed normally. If it is distributed normally, then all you need to do is know how to transform a normal distribution with mean 0 into a standard normal distribution.

For 4, you can also use the F distribution. If X1 and X2 are 2 independent RV's distributed as chi-squares with a and b degrees of freedom respectively, then (X1 / a) / (X2 / b) has the F distribution with a and b degrees of freedom.

## 1. What is statistical inference?

Statistical inference is the process of using sample data to make conclusions or predictions about a larger population. It involves using statistical methods to analyze and interpret data in order to make inferences about a population parameter.

## 2. What is the purpose of statistical inference?

The purpose of statistical inference is to use sample data to make conclusions or predictions about a larger population. It allows us to draw conclusions that can be generalized to the entire population, rather than just the specific sample that was collected.

## 3. What are the two main types of statistical inference?

The two main types of statistical inference are estimation and hypothesis testing. Estimation involves using sample data to estimate the value of a population parameter, while hypothesis testing involves using sample data to test a hypothesis about a population parameter.

## 4. What is the difference between descriptive and inferential statistics?

Descriptive statistics involve summarizing and describing data, while inferential statistics involve making inferences and conclusions about a population based on sample data. Descriptive statistics are used to describe what is happening in a dataset, while inferential statistics are used to make predictions or draw conclusions about a larger population.

## 5. What are some common methods of statistical inference?

Some common methods of statistical inference include confidence intervals, hypothesis testing, and regression analysis. These methods use sample data to make inferences about a population, and each has its own strengths and limitations depending on the type of data and research question being addressed.

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