Question
In: Statistics and Probability
Indicate whether the following statement is true or false. Both the t-distribution and standard normal distribution...
Indicate whether the following statement is true or false.
Both the t-distribution and standard normal distribution have the same mean, but the t-distribution has a smaller standard deviation than the standard normal distribution.
Solutions
Expert Solution
Answer: True.The statement is true.
The mean of the distribution equal to 0 right because it is also symmetric just like a normal distribution. The variance that is the spread is given by v divided by v minus 2, where v is the degrees of freedom and v is greater than 2, the variance is always greater than 1 so the degrees of freedom reach infinity the t-distribution reaches a standard normal distribution.
The T-distribution should not be used with small samples that are not approximately normal. In a normal distribution with a very small sample size is followed.
The standard error is a standard deviation of the sampling distribution of a set of means taken from a population.
For e.g. If I have a sample size of 4, mean is 100, the sample standard deviation is 10, then when we do a t-distribution, mean is still 100, the standard deviation will be 10 divided by 4 that is 10 by 2 which is 5.
Hence, when doing the t-distribution standard deviation is smaller than the standard normal distribution.
Note: For more Information/Properties, I have mentioned the source name and Link: Chapter 2. The Normal and t-Distributions (opentextbc)
https://opentextbc.ca/introductorybusinessstatistics/chapter/the-normal-and-t-distributions-2/#:~:text=The%20t%2Ddistribution%20can%20be,more%20about%20later%2C%20is%20calculated.
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