Question

Assume that the population of paired differences is normally distributed.

3) A test of writing ability is given to a random sample of students before and after they completed a formal
writing course. The results are given below. Construct a 99% confidence interval for the mean difference
between the before and after scores.

Before 70 80 92 99 93 97 76 63 68 71 74
After 69 79 90 96 91 95 75 64 62 64 76

(a) Find the best point estimate for μd .

(b) Find the critical values by first sketching the t distribution curve and identifying the indicated area on the
graph.

(c) Find the margin of error. Be sure to set up the equation.

(d) Construct the confidence interval.

Answer 1

Before	After	Difference
70	69	1
80	79	1
92	90	2
99	96	3
93	91	2
97	95	2
76	75	1
63	64	-1
68	62	6
71	64	7
74	76	-2

∑x = 22

∑x² = 114

n = 11

Mean , x̅_d = Ʃx/n = 22/11 = 2

Standard deviation, s_d = √[(Ʃx² - (Ʃx)²/n)/(n-1)] = √[(114-(22)²/11)/(11-1)] = 2.6458

a) point estimate for μ_d = 2

b) df = n-1 = 10

Two tailed critical value, t-crit = T.INV.2T(0.01, 10) = 3.169

Reject Ho if t < -3.169 or if t > 3.169

c) Margin of error , E = t-crit*s_d/√n = 3.169*2.6458/√11 = 2.5282

d) 99% Confidence interval :

Lower Bound = x̅_d - E = 2 - 2.5282 = -0.5282

Upper Bound = x̅_d + E = 2 + 2.5282 = 4.5282