heights from class survey			Let's see if Marlboro male students are taller than female students. Method: difference of means m1 = mean of male heights m2 = mean of femail heights H0 : males and females are the same height diff = m1-m2 = 0 Halpha: males are taller (one tail) m1 > m2 Standard deviation of m1 is sigma_m1 = 2.94 / sqrt(41) = 0.459 (This is also called the "Standard Error"). Standard deviation of m2 is sigma_m2 = = 3.42/sqrt(44) = 0.516 Standard deviation of difference is sigma_diff = sqrt( sigma_m1^2 + sigma_m2^2) = sqrt( 0.459^2 + 0.516^2) = 0.691 Let's choose alpha significance level = 0.01. Then our cutoff value for diff is where the probability of diff > cutoff is 0.01. One way to find the cutoff is to calculate the z-score, then look in our textbook table D-4, then convert back to a corresponding raw score. Another is to use Excel's NORMINV function. Type =NORMINV(0.99,0,0.691) into a cell, hit return, and get cutoff = 1.61. In other words, a normal variable with mean=0 and sigma=0.691 is bigger than 1.61 only 1% of the time. Therefore, our decision rule is to reject the null hypothesis if m1-m2 > 1.61. The actual value is diff = m1-m2 = mean male - mean female = 70.86 - 64.64 = 6.22 and so we *do* reject the null hypothesis: Male students at Marlboro are taller than female students.
								male stats
	male	female
	64	49						Mean	70.8597561
	65	60						Standard Error	0.459988134
	66	60						Median	71
	68	61						Mode	71
	68	62						Standard Deviation	2.945361173
	68	62						Sample Variance	8.675152439
	68.5	62						Kurtosis	1.37364123
	69	62						Skewness	0.339982338
	69	63						Range	15
	69	63						Minimum	64
	69	63						Maximum	79
	69.75	63						Sum	2905.25
	70	63.5						Count	41
	70	64
	70	64
	70	64						female stats
	71	64
	71	64						Mean	64.64204545
	71	64						Standard Error	0.516342191
	71	65						Median	65.25
	71	65						Mode	66
	71	65						Standard Deviation	3.42502662
	71	65.5						Sample Variance	11.73080735
	71	66						Kurtosis	9.46203557
	71	66						Skewness	-2.089331545
	72	66						Range	23
	72	66						Minimum	49
	72	66						Maximum	72
	72	66						Sum	2844.25
	72	66						Count	44
	72	66
	73	66						The histogram we drew before made the height difference very clear, if you look back. Another way to see this is with confidence intervals. Both sigma_m1 and sigma_m2 are about 0.5, so 3 sigma is about 1.5. At 3 sigma (99.5% confidence), the means are male height = 70.86 +- 1.5 female height = 64.64 +- 1.5 which clearly don't overlap.
	73	66
	73	66
	74	66.25
	74	67
	74	67
	75	67
	78	67
	79	68
	68	68
		69
		69
		72