# Independent trials, dependent condition

• caffeine
In summary, the conversation discusses the probability of two randomly selected students having an average test score higher than a given value. The approach to solving this problem involves defining a new random variable Y = X1 + X2 and calculating the probability of Y being greater than 2k. The conversation also mentions the Central Limit Theorem and how it applies to this problem. Finally, the conversation touches on the mean and standard deviation of Y and provides a link for further information.
caffeine
A friend gave me this problem; it's been years since I've taken probability, and I'm really rusty. I'm curious how to solve it.

Suppose you have a normal pdf with mean mu and stdev sigma that represents the distribution of test scores.

What's the probability that two randomly selected students will have an average higher than k?

So if I let X1 and X2 be the test scores of two randomly selected students, and if they have values of k1 and k2, then I want to know the probability that:

k1 + k2 > 2k

or

P(X1 + X2 > 2k)

I'm stumped. How does one approach a problem like that?

Thanks!

You need to define a new random variable Y = X1 + X2, then calculate P(Y > 2k). If each X is normally distributed, then Y is, too. Proof. (Even if X's were non-normal, The Central Limit Theorem states that the sum of N independent identical random variables approaches the normal distribution as N goes to infinity.)

EnumaElish said:
You need to define a new random variable Y = X1 + X2, then calculate P(Y > 2k). If each X is normally distributed, then Y is, too. Proof. (Even if X's were non-normal, The Central Limit Theorem states that the sum of N independent identical random variables approaches the normal distribution as N goes to infinity.)

OK, thanks. Is it correct that the mean of the distribution for Y is the sum of the averages (so mu' = mu + mu) and the stdev of the distribution for Y is the RMS sum of the deviations (so sigma' = sqrt{\sigma^2 + \sigma^2} = \sqrt{2}\sigma?

Thanks! I can't believe how much I've forgotten

## What is the difference between independent trials and dependent condition?

Independent trials refer to a set of experiments or events that are not affected by the outcome of previous trials. On the other hand, dependent condition refers to a situation where the outcome of one event is influenced by the outcome of a previous event.

## How are independent trials and dependent condition used in scientific research?

Independent trials are often used to test the effectiveness of a certain treatment or intervention by comparing it to a control group. Dependent condition is commonly used in studies that involve measuring changes over time, such as in longitudinal studies.

## What are some examples of independent trials and dependent condition?

An example of independent trials is a clinical trial where participants are randomly assigned to receive either a new medication or a placebo. An example of dependent condition is a study examining the relationship between smoking and lung cancer, where smoking is the independent variable and lung cancer is the dependent variable.

## How do you determine if an experiment has independent or dependent variables?

To determine if an experiment has independent or dependent variables, you need to identify what is being manipulated or changed (independent variable) and what is being measured or observed (dependent variable). If the outcome of the experiment is affected by the independent variable, it is considered a dependent condition.

## Why is it important to understand the difference between independent trials and dependent condition?

Understanding the difference between independent trials and dependent condition is crucial in designing and conducting scientific research. It allows researchers to accurately interpret their findings and draw meaningful conclusions about the relationship between variables. Additionally, it helps to ensure the validity and reliability of the results.

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