Question

For the record, this is a homework question, not a test question. Why is the t-distribution sometimes used when calculating a confidence interval? b) (2 points) In general, is the =t.inv value larger or smaller than the =norm.inv value for a given level of confidence. c) (4 points) How do t and z compare as the sample size increases? Be sure to include a graph in your answer. Your graph should include t with a small sample size, t with a large sample size, and z.

Answer 1

1)

When the sample size is large i.e. , we use the z-distribution to calculate the confidence interval because, for larger sample size, the sampling distribution becomes normal (using the central limit theorem such that whatever the distribution of population, sample will be normally distributed) and if sample size is small i.e. , the central limit theorem doesn't employ for the sampling distribution hence population distribution affect the sampling distribution.

2)

For a given level of confidence, =T.INV value larger than the =NORM.S.INV

For example, for 95% confidence interval,

NORM.S.INV(0.975)	1.959964
T.INV.2T(0.05,100)	1.983972

4)

As the sample size increses, T.INV move towards NORM.INV value.

z value	NORM.S.INV(0.975)	1.959964
t value	T.INV.2T(0.05,5)	2.228139
	T.INV.2T(0.05,10)	1.983972
	T.INV.2T(0.05,20)	1.962339
	T.INV.2T(0.05,100)	1.960201

The following figure displays the t distribution for different sample size,