# When to Use t-distribution?

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#1
I know t-distribution is primarily used when a small sample is taken from a normally distributed population. The population's mean and variance are unknown. The sample is used to estimate mean of the population using confidence intervals.

But what if population's variance is known. Do I still use t-distribution to find confidence interval or do I use normal distribution?
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#2
If I am correct, for a small sample of a normally distributed population where population variance is known, normal distribution must be used to find confidence interval of the mean.

Also if sample is large and population variance is unknown, use normal distribution and estimate variance using unbiased variance formula.

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I am guessing that the same ideas which apply when finding confidence intervals apply to finding P(X < c) for some numeric value of c (X from a normally distributed population but it can be estimated using either normal or t-distribution).
Last edited by esrever; 2 years ago
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2 years ago
#3
If the population variance is known, then the population standard deviation is known.
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#4
(Original post by NotNotBatman)
If the population variance is known, then the population standard deviation is known.
So what does that imply?
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2 years ago
#5
(Original post by esrever)
If I am correct, for a small sample of a normally distributed population where population variance is known, normal distribution must be used to find confidence interval of the mean.

Also if sample is large and population variance is unknown, use normal distribution and estimate variance using unbiased variance formula.
Yes; the point here is whether the population variance is known, or whetehr it is estimated from the sample. If you know it, the normal distribution is used, if you have to estimate it, the t-distribution is used. If the sample is large, then the normal is a very good approximation to the corresponding t-distribution, so we tend to use the normal for simplicity's sake.

Just a little dirty secret that we illuminati keep from you: the square root of the unbiased estimator of the population variance is an estimator of the population standard deviation - but it is a biased estimator! The bias tends to be small, so we ignore this.
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#6
(Original post by Gregorius)
Yes; the point here is whether the population variance is known, or whetehr it is estimated from the sample. If you know it, the normal distribution is used, if you have to estimate it, the t-distribution is used. If the sample is large, then the normal is a very good approximation to the corresponding t-distribution, so we tend to use the normal for simplicity's sake.

Just a little dirty secret that we illuminati keep from you: the square root of the unbiased estimator of the population variance is an estimator of the population standard deviation - but it is a biased estimator! The bias tends to be small, so we ignore this.
Thanks for help Spoiler:
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ILLUMINATI CONFIRMED!!
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#7
(Original post by Gregorius)
Yes; the point here is whether the population variance is known, or whetehr it is estimated from the sample. If you know it, the normal distribution is used, if you have to estimate it, the t-distribution is used. If the sample is large, then the normal is a very good approximation to the corresponding t-distribution, so we tend to use the normal for simplicity's sake.

Just a little dirty secret that we illuminati keep from you: the square root of the unbiased estimator of the population variance is an estimator of the population standard deviation - but it is a biased estimator! The bias tends to be small, so we ignore this.
Is this because a small sample doesn't estimate population variance well from sample variance and that's why t-distribution is used to reduce error?
Last edited by esrever; 2 years ago
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2 years ago
#8
(Original post by esrever)
Is this because a small sample doesn't estimate population variance well from sample variance and that's why t-distribution is used to reduce error?
Well, it's simply because of the fact that you're estimating the variance from the sample, rather than knowing it as a fixed constant. A small sample will estimate the variance with less precision than a large one; but strictly speaking, small sample or large you have to take the fact that an extra parameter is subject to uncertainty into account.
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#9
(Original post by Gregorius)
Well, it's simply because of the fact that you're estimating the variance from the sample, rather than knowing it as a fixed constant. A small sample will estimate the variance with less precision than a large one; but strictly speaking, small sample or large you have to take the fact that an extra parameter is subject to uncertainty into account.
Thank you so much 0
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