Question

The ballistic coefficient is a measure of body’s ability to overcome air resistance in flight. That parameter is inversely proportional to the deceleration of a flying body and is very important for bullets. The ballistic coefficient was measured for the bullets of two versions of 9 mm Makarov cartridges, PM and PMM (which is a later and modified version). Sample bullets are chosen randomly.

PM	12.93	12.89	13.13	13.11	12.81	12.83	13.11	12.67	12.85	12.99	13.05	12.75	13.08	13.17	13.16	12.64
PMM	13.9	13.95	13.98	13.67	13.99	13.72	13.56	13.99	13.78	13.54	14.00	13.63	13.67	13.98

Use α=0.025.

Find a 95% two-sided confidence interval for the mean difference in the ballistic coefficient.

Round your answers to three decimal places (e.g. 98.765).

______ ≤μ1-μ2≤ ______

Answer 1

From the Data:

= 12.948, s₁ = 0.1765, n₁ = 16,

= 13.811, s₂ = 0.1759, n₂ = 14,

Since s₁/s₂ = 0.1765 / 0.1759 = 1.00 (it lies between 0.5 and 2) we used the pooled standard deviation

The Pooled Variance is given by:

df = n₁ + n₂ – 2 = 16 + 14 – 2 = 28

The tcritical (2 tail) for = 0.05, df = 285 is 2.0555

The Confidence Interval is given by ME, where

( ) = 12.948 - 13.811 = 0.863

The Lower Limit = -0.863 - 0.136 = -0.999

The Upper Limit = -0.863 + 0.136 = -0.727

The Confidence Interval is -0.999< µ₁- µ₂ < -0.727

_______________________________

Calculation for the mean and standard deviation:

Mean = Sum of observation / Total Observations

Standard deviation = SQRT(Variance)

Variance = Sum Of Squares (SS) / n - 1, where SS = SUM(X - Mean)²

#	PM	Mean	(X - mean )2		PPM	Mean	(X - mean )2
1	12.93	12.948	0.000324		13.9	13.811	0.007921
2	12.89	12.948	0.003364		13.95	13.811	0.019321
3	13.13	12.948	0.033124		13.98	13.811	0.028561
4	13.11	12.948	0.026244		13.67	13.811	0.019881
5	12.81	12.948	0.019044		13.99	13.811	0.032041
6	12.83	12.948	0.013924		13.72	13.811	0.008281
7	13.11	12.948	0.026244		13.56	13.811	0.063001
8	12.67	12.948	0.077284		13.99	13.811	0.032041
9	12.85	12.948	0.009604		13.78	13.811	0.000961
10	12.99	12.948	0.001764		13.54	13.811	0.073441
11	13.05	12.948	0.010404		14	13.811	0.035721
12	12.75	12.948	0.039204		13.63	13.811	0.032761
13	13.08	12.948	0.017424		13.67	13.811	0.019881
14	13.17	12.948	0.049284		13.98	13.811	0.028561
15	13.16	12.948	0.044944
16	12.64	12.948	0.094864

PM		PPM
n1	16	n2	14
Sum	207.17	Sum	193.36
Mean	12.948	Mean	13.811
SS	0.467	SS	0.402374
Variance	0.0311	Variance	0.03095
SD - PM	0.1765	SD - PPM	0.1759

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(x1 - x2)

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