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Why we use t-distribution when sample size is small? (n

What is the reason that Student-t Distribution is used when the number of samples is smallHow is it that the required sample size for a specified error and confidence is not dependent on population size?How do I use student's-t distribution without the sample size?Sampling distribution with large sample sizeWhen the population variance is unknown, we should use t-distribution.Should I use the t distribution instead of normal for large samples?What is the standard deviation of the sample distribution for the sample standard deviation?Why prefer the t-score when the sample size is low?Is a t distribution for a certain degree of freedom equivalent to the sample mean distribution for the corresponding sample size?Why do we use the t-distribution for this problem? Isn't the sample size too large?Sample Size Proof

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When population is normal, distribution of sample mean is also normal regardless of sample size n. Then, why we use t-distribution instead of normal distribution when sample size is small? (n < 30)

statistics normal-distribution

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asked Mar 29 at 11:06

firia2000firia2000

1011

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  But you have an unknown standard deviation I'm guessing. When this is the case and you replace the standard deviation with the sample standard deviation, you end up with a $t$-distribution. (Remember if $X_i$ are iid $Nleft(mu,sigma^2right)$, $overlineX_n =frac1nsumlimits _i =1^nX_i$ and $S = sqrtfrac1n-1sumlimits_i=1^nleft(X_i- overlineX_nright)^2$, then $fracoverlineX_n - muS/sqrtnsim t_n-1$.)
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  – Minus One-Twelfth
  Mar 29 at 11:10
  
  

  
  
  
  


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  When population variance $sigma^2$ is unknown and estimated by sample variance $S^2,$ you should always use a t test for most accurate results. For $n > 30$ results from an incorrect z test may be a serviceable approximation. The approx 'rule of 30' works only for 95% CIs or 5% significance levels because 1.96 cuts 2.5% from upper tail of std normal and for $n > 30$ numbers around 2.0 cut 2.5% from upper tail of Student's t dist'n with $n - 1 ge 30$ degrees of freedom.
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  – BruceET
  Mar 29 at 16:14
  
  

  
  
  
  


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  See math.stackexchange.com/questions/2992333
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  – cgiovanardi
  Mar 29 at 22:09
  

  
  
  
  

add a comment | 

0

$begingroup$

When population is normal, distribution of sample mean is also normal regardless of sample size n. Then, why we use t-distribution instead of normal distribution when sample size is small? (n < 30)

statistics normal-distribution

share|cite|improve this question

asked Mar 29 at 11:06

firia2000firia2000

1011

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  But you have an unknown standard deviation I'm guessing. When this is the case and you replace the standard deviation with the sample standard deviation, you end up with a $t$-distribution. (Remember if $X_i$ are iid $Nleft(mu,sigma^2right)$, $overlineX_n =frac1nsumlimits _i =1^nX_i$ and $S = sqrtfrac1n-1sumlimits_i=1^nleft(X_i- overlineX_nright)^2$, then $fracoverlineX_n - muS/sqrtnsim t_n-1$.)
  $endgroup$
  – Minus One-Twelfth
  Mar 29 at 11:10
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  When population variance $sigma^2$ is unknown and estimated by sample variance $S^2,$ you should always use a t test for most accurate results. For $n > 30$ results from an incorrect z test may be a serviceable approximation. The approx 'rule of 30' works only for 95% CIs or 5% significance levels because 1.96 cuts 2.5% from upper tail of std normal and for $n > 30$ numbers around 2.0 cut 2.5% from upper tail of Student's t dist'n with $n - 1 ge 30$ degrees of freedom.
  $endgroup$
  – BruceET
  Mar 29 at 16:14
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See math.stackexchange.com/questions/2992333
  $endgroup$
  – cgiovanardi
  Mar 29 at 22:09
  

  
  
  
  

add a comment | 

$begingroup$

When population is normal, distribution of sample mean is also normal regardless of sample size n. Then, why we use t-distribution instead of normal distribution when sample size is small? (n < 30)

statistics normal-distribution

share|cite|improve this question

asked Mar 29 at 11:06

firia2000firia2000

1011

$endgroup$

share|cite|improve this question

asked Mar 29 at 11:06

firia2000firia2000

1011

share|cite|improve this question

asked Mar 29 at 11:06

firia2000firia2000

1011

asked Mar 29 at 11:06

firia2000firia2000

1011

asked Mar 29 at 11:06

firia2000firia2000

1011