Question

In: Statistics and Probability

Graded problem #2. The table below gives the cost per unit of a manufactured piece as...

Graded problem #2. The table below gives the cost per unit of a manufactured piece as a function of the number of units produced. Take the number of units produced as the explanatory variable, x, and the cost per unit as the response variable y.

Units 10 20 50 100 150 200

Cost 140 155 153 150 180 160

a. Predict the cost per unit when 80 units are produced

b. Find a 90% confidence interval for the slope b1

c. Find a 90% confidence interval for the predicted value you found in answer b.

Expert Solution

	X	Y	X * Y	X²	Ŷ	Sxx =Σ (Xi - X̅ )²	Syy = Σ( Yi - Y̅ )²	Sxy = Σ (Xi - X̅ ) * (Yi - Y̅)
	10	140	1400	100.0000	147.339	6136.106	266.777	1279.441
	20	155	3100	400.0000	148.488	4669.440	1.778	91.109
	50	153	7650	2500	151.932	1469.442	11.111	127.776
	100	150	15000	10000	157.673	136.112	40.111	-73.889
	150	180	27000	22500	163.414	3802.782	560.113	1459.447
	200	160	32000	40000	169.155	12469.452	13.445	409.448
Total	530	938	86150	75500	672.000	28683.333	893.333	3293.333

X̅ = Σ (Xi / n ) = 530/6 = 88.3333
Y̅ = Σ (Yi / n ) = 938/6 = 156.3333

Equation of regression line is Ŷ = a + bX
b = ( n Σ(XY) - (ΣX* ΣY) ) / ( n Σ X² - (ΣX)² )
b = ( 6 * 86150 - 530 * 938 ) / ( 6 * 75500 - ( 530 )²)
b = 0.1148

a =( ΣY - ( b * ΣX ) ) / n
a =( 938 - ( 0.1148 * 530 ) ) / 6
a = 146.1912
Equation of regression line becomes Ŷ = 146.1912 + 0.1148 X

Part a)

= 146.1912 + 0.1148 X
= 155.3752

Estimated Error Variance (σ̂²) =
S² = ( 893.3333 - 0.1148 * 3293.3333 ) / 6 - 2
S² = 128.814659
S = 11.3497

Part b)

Confidence Interval

= 2.132 ( Critical value from t table )

90% confidence interval is -0.0281 < < 0.2577

Part c)

Confidence Interval of
= 146.1912 + 0.1148 X
= 155.3752

= 2.132 ( From t table )
X̅ = (Xi / n ) = 530/6 = 88.3333
= 155.3752

90% confidence interval is ( 145.4252 < < 165.3252 )

orchestra answered 1 year ago

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