2 questions probability and statistics

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Question 1

  1. The article “An Investigation of the CaCO3-CaF2-K2SiO3-SiO2-Fe Flux System Using the Submerged Arc Welding Process on HSLA-100 and AISI-1081 Steels” (G. Fredrickson, M.S. Thesis, Colorado School of Mines, 1992) describes an experiment in which welding fluxes with differing chemical compositions were prepared. Several welds using each flux were made on AISI-1018 steel base metal.The results of hardness measurements, on the Brinell scale, of five welds using each of four fluxes are presented in Table below. Assume equal variances and level of significance α=0.05. Hint: only one factor i.e. flux type is considered. Use 4DP for all your answers.

    Flux Sample Values Sample Mean Sample Standard Deviation
    A 250 264 256 260 239 253.8 9.7570
    B 263 254 267 265 267 263.2 5.4037
    C 257 279 269 273 277 271.0 8.7178
    D 253 258 262 264 273 262.0 7.4498

    H0 and H1 are stated as: H0: μ1= μ2= μ3= μ4 H1: not all µj are equal

    1. SSA = , SSW =
    2. MSA = ,MSW =
    3. F test statistic Fstat = .
    4. Critical value Fα,df1,df2 =
    5. p-value= (You may use R or data analysis tool in Excel to find the p-value)
    6. Decision: we (fill in the blank by “should” or “should not“) reject the null hypothesis.

      Question 2

      1. A study is made of the effect of curing temperature on the compressive strength of a certain type of concrete. Five concrete specimens are cured at each of four temperatures, and the compressive strength of each specimen is measured(in MPa). The results are as follows:

        Temperature (C0) Strengths (MPa)
        0 31.2 29.6 30.8 30.0 31.4
        10 30.0 27.7 31.1 31.3 30.6
        20 35.9 36.8 35.0 34.6 36.5
        30 38.3 37.0 37.5 36.1 38.4

        Assume equal variances and α=0.05. ANOVA analysis may be performed to check if the mean strength differs with temperature.
        Use 4DP for all your solutions H
        0 and H
        1 are stated as: H0: μ
        1= μ
        2= μ
        3= μ
        4 H1:
        not all µ
        j are equal

        1. SSA = , SSW =
        2. MSA = , MSW =
        3. F test statistic Fstat=.
        4. Critical value F α,df1,df2 =
        5. Decision: we (fill in the blank by “should” or “should not“) reject the null hypothesis.
 
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