Question

An article in Environment International (Vol. 18, No. 4, 1992) describes an experiment in which the amount of radon released in showers was investigated. Radon-enriched water was used in the experiment, and six different orifice diameters were tested in shower heads. The data from the experiment are shown in the following table.

13.2.1 GO Tutorial An article in Environment International (Vol. 18, No. 4, 1992) describes an experiment in which the amount of radon released in showers was investigated. Radon-enriched water was used in the experiment, and six different orifice diameters were tested in shower heads. The data from the experiment are shown in the following table. Orifice Diameter Radon Released (%) 0.37 80 0.51 75 8483 85 75 7979 73 76 77 72 74 74 0.71 74 1.02 67 1.40 62 62 68 69 1.99 60 61 64 66 (a) Does the size of the orifice affect the mean percentage of radon released? Use a = 0.05. Round all intermediate calculations to four decimal places (e.g. 12.3456). Since For in the rejection region, there sufficient evidence to conclude that the mean percentage varies according to orifice diameter (b) Find bounds on the P-value for the F-statistic in part (a). Use only Table VI of Appendix A. Statistical Tables and Charts the absolute tolerance is +/-0.05

Answer 1

Consider 0:37 رد 051 o f = orific diameter is Y2 = Yg= Y4= Ys= Yo = ) 1.02 1.40 1.99 Yij groups i=1,2 where i=lRnb j= absn j = 1,2,3,4 ne 6x4 = 24 ..-Elyij K= 6 26 sum of beta group = s.56 = E£(7;.-J.) Whaw 9 ;= y is.s.b. 1139.7 square within groups =s.sw. =SS EECYij-Ji.) 55-6.5 138.3 Sum of square total = SS.T= ssib tag:w. Sum of 2 ss. We CamScanner

SIST= 1139.7+ 158.3 S.S.T= 12 #8 direcbet” gooups)= kg=61es dif. (withing roups) def. (Total) 6-1-6-1-5 = neler 24-6-18 2-1 - 26-1 =23 M.S.5-6 - Sisib 1139.7 227.942 M.sis.w. $5.0 n-k 1383 18 7.68) meems -L Ho - Mi=2=Mzale = 154 not all equal. Test statistic & under rio mis sibi Fo Colfo Misisiw, 227.942 F-68) fo = 29.68 critical F= F-1,6723,5 C5, 1B) 0.03 coitican F= 2.77285 cusing Table CS'Scanned with or software) CamScanner

Hore . Decision Rule :- peject to If Cal Fo > Tab F ' Fo= 29.68]> [coitical [coitical F= 2:7428 Test statistic in the Rejection Region there is sufficient evietence to Conclude that the mom percentage according to to orifice diameter PC F2 Cal Fo) FPCF 29.68) = I-PC FC 29.68) - curing varies ac p-rahe = 1 1 softwares - -vawe p-val CS Scanned with CamScanner

9 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 Critical values of F for the 0.05 si ficanc level: 1 2 3 4 5 6 161.45 199.50 215.71 224.58 20.16 $3.99 236.77 18.51 19.00 19.16 19.25 19.33 19.35 10.13 9.55 9.28 9.12 9.01 8.94 8.89 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.61 5.79 5.41 5.19 5.05 4.95 4.88 5.99 5.14 4.76 4.53 4.39 4.28 4.21 5.59 4.74 4.35 4.12 3.97 3.87 3.79 5.32 4.46 4.07 3.84 3.69 3.58 3.50 5.12 4.26 3.86 3.63 3.48 3.37 3.29 4.97 4.10 3.71 3.48 3.33 3.22 3.14 4.84 3.98 3.59 3.36 3.20 3.10 3.01 4.75 3.89 3.49 3.26 3.11 3.00 2.91 4.67 3.81 3.41 3.18 3.03 2.92 2.83 4.60 3.74 3.34 3.11 2.96 2.85 2.76 4.54 3.68 3.29 3.06 2.90 2.79 2.71 4.49 3.63 3.24 3.01 2.85 2.74 2.66 4.45 3.59 3.20 2.97 2.70 2.61 4.41 3.56 3.16 2.93 2.77 2.66 2.58 4.38 3.52 3.13 2.90 2.63 2.54 4.35 3.49 3.10 2.87 2.71 2.60 2.51 4.33 3.47 3.07 2.84 2.69 2.57 2.49 4.30 3.44 3.05 2.82 2.66 2.55 2.46 4.28 3.42 3.03 2.80 2.64 2.53 2.44 4.26 3.40 3.01 2.78 2.62 2.51 2.42 4.24 3.39 2.99 2.76 2.60 2.49 2.41 4.23 3.37 2.98 2.74 2.59 2.47 2.39 4.21 3.35 2.96 2.73 2.57 2.46 2.37 4.20 3.34 2.95 2.71 2.56 2.45 2.36 4.18 3.33 2.93 2.70 2.55 2.43 2.35 4.17 3.32 2.92 2.69 2.53 2.42 2.33 4.16 3.31 2.91 2.68 2.52 2.41 2.32 4.15 3.30 2.90 2.67 2.51 2.40 2.31 4.14 3.29 2.89 2.66 2.50 2.39 2.30 4.13 3.28 2.88 2.65 2.49 2.38 2.29 4.12 3.27 2.87 2.64 2.49 2.37 2.29 8 238.88 19.37 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 2.42 2.40 2.38 2.36 2.34 2.32 2.31 2.29 2.28 2.27 2.26 2.24 2.24 2.23 2.22 10 241.88 19.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 2.32 2.30 2.28 2.26 2.24 2.22 2.20 2.19 2.18 2.17 2.15 2.14 2.13 2.12 2.11 240.54 19.39 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 2.37 2.34 2.32 2.30 2.28 2.27 2.25 2.24 2.22 2.21 2.20 2.19 2.18 2.17 2.16 18 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Minitab - Untitled - Eile Edit Data Calc Stat Graph Editor Tools Window Help Assistant TOHO % *SOMO 2 BO GF -- ASK 2 h+Y TOO Session X 7/30/2020 7:18:00 AM Welcome to Minitab, press F1 for help. One-way ANOVA: y1, y2, y3, y4, y5, y6 Method Null hypothesis Alternative hypothesis Significance level All means are equal At least one mean is different a = 0.05 Equal variances were assumed for the analysis. Factor Information Factor Factor Levels 6 Values yi, y2, y3, y4, y5, y6 Analysis of Variance Source Factor Error Total F-Value 29.68 DF 5 18 23 P-Value 0.000 Seq SS 1139.7 138.3 1278.0 Adj ss 1139.7 138.3 Contribution 89.18% 10.82% 100.00% Adj MS 227.942 7.681 Model Summary R-sa 89.18% R-sq (adj) 86.18% PRESS 245.778 R-sq (pred) 80.77% 2.77138 Means Factor y1 y2 N 4 4 Mean 83.00 77.00 St Dev 2.16 2.31 95% CI ( 80.09, 85.91) ( 74.09, 79.91) Current Worksheet: Worksheet 1 Type here to search IT e Editable 7:42 AM 7/30/2020 1 R 3

Minitab - Untitled - Eile Edit Data Calc Stat Graph Editor Tools Window Help Assistant TOHO % *SOMO 2018 fx -- AMX2 W+Y WITO OOM Session X Total 23 1278.0 100.00% Model Summary S 2.77138 R-sa 89.188 R-sq(adj) 86.18% PRESS 245.778 R-sq (pred) 80.77% Means N 4 4 Factor yl y2 y3 y4 y5 y6 4 Mean 83.00 77.00 75.000 71.75 65.25 62.75 St Dev 2.16 2.31 1.826 3.30 3.77 2.75 95% CI ( 80.09, 85.91) ( 74.09, 79.91) (72.089, 77.911) ( 68.84, 74.66) ( 62.34, 68.16) (59.84, 65.66) 4 4 4 Pooled StDev = 2.77138 Interval Plot of y1, y2, ... Inverse Cumulative Distribution Function F distribution with 5 DF in numerator and 18 DF in denominator P(Xsx) 0.95 x 2.77285 Cumulative Distribution Function F distribution with 5 DF in numerator and 18 DF in denominator P ( X S x) 1.00000 29.68 Current Worksheet: Worksheet 1 Type here to search 0 Editable 7:42 AM 7/30/2020 IT e 1 R 3