Question

A machine produces metal rods used in an automobile suspension system. A random sample of n = 12 rods is selected, and their diameters are measured. The resulting data in millimeters are shown here: 8.23 8.31 8.42 8.29 8.19 8.24 8.19 8.29 8.30 8.14 8.32 8.40

a) Calculate the sample mean x bar and the unbiased variance estimate s^2

b) Let α = 0.05. Determine the percentage point tα/2, n−1 of the corresponding t-distribution.

c) Assuming that the data comes from a normal population N (µ, σ^2 ), with unknown mean µ and unknown variance σ^2 , find a 95% two-sided confidence interval on mean value µ of the rod diameter.

d) Test the null hypothesis H0 : µ = µ0 = 8.20 mm,

Answer 1

a)

Mean = = (Sum of all observations)/(total number of observations)

unbiased variance estimate s^2 =

standard deviation s =

Sl.No.	Data (Xi)	(Xi - bar(X))^2
1	8.23	0.0025
2	8.31	0.0009
3	8.42	0.0196
4	8.29	1E-04
5	8.19	0.0081
6	8.24	0.0016
7	8.19	0.0081
8	8.29	1E-04
9	8.3	0.0004
10	8.14	0.0196
11	8.32	0.0016
12	8.4	0.0144
Total	99.32	0.077
Average	8.277
Variance	0.007
Standard deviation	0.084

sample mean x bar = 8.277

and the unbiased variance estimate s^2 = 0.007

b) Let α = 0.05. Determine the percentage point tα/2, n−1 of the corresponding t-distribution.

t0.025,11 = 2.201

c) 95% CI =

95% CI = ( ) = ( 8.224 , 8.330 )

d) hypothesis

H0 : µ = µ0 = 8.20 mm,

H1 :

test statistic t = = = 3.175

Since the |test statistic| (3.175) is greater than |critical value| (2.201), we reject H0 and there is a significant evidence to conclude that the mean diameter of the rod is different from 8.20 mm.