Question

A new process for producing small precision parts is being studied. The process consists of mixing fine metal powder with a plastic binder, injecting the mixture into a mold, and then removing the binder with a solvent. These data are obtained on parts that should have a 1-inch diameter and whose standard deviation should not exceed 0.0025 inch. Perform all hypothesis tests at a 5% level of significance.

1.0030     0.9997     0.9990    1.0054     0.9991

1.0041     0.9988     1.0026     1.0032     0.9943

1.0021     1.0028     1.0002     0.9984     0.9999

What is the sample mean? (Round off to 5 decimal figures)

What is the sample standard deviation?

Formulate H1 for testing μ.

What is/are the critical region(s)?

What is the computed value of the test statistic?

Formulate H1 for testing for σ²

What is/are the critical region (s)?

What is the computed value of the test statistic?

                    What is the probability that the sample mean is at least 0.9988 if μ is known to be 1.0008  

                            and σ is unknown?

If σ is known to be 0.0022, find the probability that the sample standard deviation is at

                           most 0.0026?

Answer 1

parts. There is a sample of 15 The mean is given by : - Sym of all the sample values = 1.00304 1.0041+ 1.0021 +0.9997 +0.9988 +1.0028 +0.9990+1.00 26 + 1.0002.+ 1.00544 1.0032 +0.9984 + +0.9991 +0.9943 +0.9999 2 mean = 15.0126 = 1.00084 = mean | Is e = Sample standard deviation is given by : &s= t ² (xi - x)² - Telf.00216)+ (0.00326) 7 (0.00126) 7.6.00114) = 1549 (0.00201)+ (.00196) (-.00184)+ (00176) 1 + (-.00064) + (0.00456) + (.00236)2 + (-.00244) + I +6.00 174) ²7 (.00654)2 + (-.00094) 2 2 2 0.000112 0.0000 0 74 15 [os= 0.002727 Testing of hypothesis & for testing mean null hypothesis Ho: le=1. Alternative hyp. Hi : 1171 To so se pretvore Pro + : 7 lo Test statistics o REDMI NOTE 6 PRO MI DUAL CAMERA

t= 1.00084-1 D.00272/514 = (0.00084) x 3.741 0.00272 [t= 1.1553 ) for 14 defo t-tabulated value for 14 def ttab = 2.15 for 14 dof at 5% level of significance. critical region is - P [1H 7, 2.15] =.05 o ttab > teal so we accept our null hypothesis at 5% of level of significance. for standard deviation. For Variance (2) null hypothesis Ho: 02466.0025) 2 Alternative hopp. Mi: 02 76.002) Test statistics is = 22. ms - E(X:-->) = 15 X 10.00 272)² (0.0025) o 2 REDMI NOTE PRO 56 Į with 14 dif. MI DUAL CAMERA

Tabulated value [x2 ARRRR = 23.685 > Mato at 5% level of significance accept Ho and a² f10.0025)? - » 7 Probability of sample mean is at least 0.9988 when = 1.0008 and o is unknown. iep (x 0.9988) =P[(x-4) > (6.9988 - 1.0008] - 0.002 0.00272 = P[t7 -0.7352] (now take this value from the standard normal table) - o= 0.0022 prob. that sample stemdard deviation is at most 0.002 6. i we know that SNN SNN (0,0022, (0.0022 al (ocs < 0.0026) = P(-0.0022 < sar < 0.0004) EP (0.0022 </s-o i 0.0004) 10.0004 Voz 0.000y 2x15 OO REDMI NOTE 6 PRO MI DUAL CAMERA 0.0004

P(-505 < < I I = ploca <5.5) + Ploc Z<1) = 0.50+ 0.3413 = 0.8413