Question

7. Usually, anything over 30 samples is considered a large sample set however sometimes other values could be selected as the cutoff for a large sample set. Use Excel to calculate the error between the z value and Student's-t values (two-tails). Assuming that Student's-t values are the true values and the desired level of confidence is 89.6% compute the error for: a) n 20, b) n 30 and c)n 100 (Be sure to list the Excel formulas you use.) C

Answer 1

Level of confidence = 89.6 % implies probability = 0.896 at that z value or student's-t value.

Following is the excel output:

n	Prob	Z-value	Student's-t value	Error	Error %
20	0.896	1.625763	1.707596302	0.081833	4.792287
30	0.896	1.625763	1.678476052	0.052713	3.1405075
100	0.896	1.625763	1.640859592	0.015096	0.9200182

Formulas used:

For z-value we calculate 'z' such that Prob(-z<X<z) = 0.896 using following formula:

Z = NORM.S.INV(cumulative probability at positive z = (1+Prob)/2)

Similarly t-value we use inverse function of 2 tailed t-distribution:

t-value = T.INV.2T(probability of tails = 1 - Prob, deg of freedom = n-1)

One can easily note that as 'n' increases error in z-value w.r.t t-value decreases.

Following is excel output with formulas:

n	Prob	Z-value	Student's-t value	Error	Error %
20	0.896	=NORM.S.INV((1+C3)/2)	=T.INV.2T(1-C3,B3-1)	=ABS(E3-D3)	=100*F3/E3
30	0.896	=NORM.S.INV((1+C4)/2)	=T.INV.2T(1-C4,B4-1)	=ABS(E4-D4)	=100*F4/E4
100	0.896	=NORM.S.INV((1+C5)/2)	=T.INV.2T(1-C5,B5-1)	=ABS(E5-D5)	=100*F5/E5