Which of the following statements is true about the t-distribution with k degrees of freedom?
i. the t-distribution is symmetric
ii. the t-distribution with k degrees of freedom has a smaller variance than the t-distribution with k+1 degrees of freedom
iii. the t-distribution has a larger variance than the standard normal (z) distribution
any one, or all three could be right but i don't know which ones are
The first and third are true. The t-distribution is symmetrical; the distribution is lower and the tails are wider than the normal distribution. I think the second one should be k-1 to make the statement true, but check my thinking on that one.
I hope this will help.
you're right
Well, you're partly right! The first statement is true - the t-distribution is indeed symmetric.
However, the second statement is false. The t-distribution with k degrees of freedom actually has a larger variance than the t-distribution with k+1 degrees of freedom. As the degrees of freedom increase, the t-distribution becomes closer to the standard normal distribution.
As for the third statement, it is also false. The t-distribution actually has a smaller variance than the standard normal (z) distribution. The t-distribution accounts for uncertainty due to small sample sizes, while the standard normal distribution assumes large sample sizes.
So in conclusion, statement i is true, while statements ii and iii are false.
Hope that clears things up! Feel free to ask more questions.
You are correct! The first statement is true. The t-distribution is symmetric, meaning it is centered at zero and has equal probabilities on both sides of the distribution.
The third statement is also true. The t-distribution has a larger variance than the standard normal (z) distribution. This is because the t-distribution has more variability due to the smaller sample size (k degrees of freedom) compared to the larger sample size (infinite degrees of freedom) assumed in the standard normal distribution.
However, the second statement is not true. The t-distribution with k degrees of freedom has a larger variance than the t-distribution with k+1 degrees of freedom. As the degrees of freedom increase in the t-distribution, the distribution becomes closer to the standard normal distribution, resulting in a smaller variance.
So the correct statements are:
i. The t-distribution is symmetric.
iii. The t-distribution has a larger variance than the standard normal (z) distribution.
I apologize for the confusion in my initial response. Thank you for pointing it out!
To determine the correct statements about the t-distribution with k degrees of freedom, let's analyze each statement one by one:
i. The t-distribution is symmetric.
This statement is indeed true. The t-distribution is symmetric around its mean value, just like the standard normal distribution. This means that the probabilities of observing values on the left and right tails of the t-distribution are equal.
ii. The t-distribution with k degrees of freedom has a smaller variance than the t-distribution with k+1 degrees of freedom.
This statement is not true. The variance of the t-distribution increases as the degrees of freedom (k) increase. In other words, as the sample size (k) increases, the t-distribution becomes more concentrated around its mean and resembles the standard normal distribution more closely. Therefore, the t-distribution with k+1 degrees of freedom will have a smaller variance than the t-distribution with k degrees of freedom.
iii. The t-distribution has a larger variance than the standard normal (z) distribution.
This statement is also true. The t-distribution has a larger variance than the standard normal distribution, regardless of the degrees of freedom. The larger variance of the t-distribution is one of the reasons why its tails are wider compared to the standard normal distribution.
In summary, statement i) is true, statement ii) is false, and statement iii) is true.