The Goodman Tire and Rubber Company periodically tests its tires for tread wear under simulated road conditions. To study and control the manufacturing process, 20 samples, each containing three radial tires, were chosen from different shifts over several days of operation; the data collected are shown below. Assuming that these data were collected when the manufacturing process was believed to be operating in control, develop the R and charts. R Chart: (to 2 decimals)
UCL =
LCL =
Chart: (to 1 decimal)
UCL =
LCL =

Range:

UCL = 4.73

LCL = 18.08

MEAN :

UCL = 27.115

LCL = 31.219

Step-by-step explanation:

Given the data:

The mean and range of each sample :

Sample __ Thread wear __ xbar __ R

1 ___31 __ 42 ___ 28 33.67 _14

2___ 26 _ 18 35 26.33 _17

3___25 __30 ___ 3429.67 _ 9

4 __ 17 __ 25 ___ 21 21 ___ 8

5 __ 38 _ 29 ___ 35 34 __ 9

6 __ 41 __42 ___36 39.67_ 6

7 __ 21 __ 17 ___29 22.33 _12

8 __ 32 __26___28 28.67 _ 6

9 __ 41 __ 34 __ 33 36 __8

10__29___17___30 25.33_ 13

11 __26 __ 31 __ 40 32.33_ 14

12__23 __ 19 __ 45 12.33 __6

13 __17 __ 24 __ 3224.33__15

14 __43__ 35___17 31.67 _ 26

15__18 ___25__ 29 24 ___ 11

16__30___42___31 34.33__ 12

17__28___36 __ 32 32 8

18__40 __ 29 __ 31 33.33 __ 11

19__18 ___29__ 28 25 11

20_ 22 __ 34 __ 26 ___ 27.33 __12

Size per sample, sample size, n = 3

Number of samples, k = 20

We calculate the sample mean and range average :

Sample mean, x-- = Σxbar/n = 29.167

Range average, Rbar = ΣR/n = 11.4

The mean control limit :

x-- ± A2Rbar

From the x chart ;

A2 for n = 20 is A2 = 0.180

29.167 ± 0.180(11.40)

LCL = 29.167 - 0.180(11.40) = 27.115

UCL = 29.167 + 0.180(11.40) = 31.219

The Range control limit :

Rbar(1 ± 3(d3/d2))

From the R-chart :

d2 at n = 20 ; d2 = 3.735

d3 at n = 20 ; d3 = 0.729

LCL = 11.40(1 - 3(0.729/3.735)) = 4.725

UCL = 11.40(1 + 3(0.729/3.735)) = 18.075

example 2 – simplify: step 1: completely factor both the numerators and denominators of all fractions. step 2: change the division sign to a multiplication sign and flip (or reciprocate) the fraction after the division sign; essential you need to multiply by the reciprocal. step 3 : cancel or reduce the fractions.

y=3x+4

step-by-step explanation:

given the points a(0,4) and b(-4, -8)

eqn of line ab: (y-4)/(x-0)=(-8-4)/(-4-0)

(y-4)/x=-12/-4=3

y=3x+4