Question #341575

A manufacturer claims that the average tensile

strength of thread A exceeds the average tensile

strength of thread B. To test this claim, 50 pieces of

each type of thread are tested under similar

conditions. Type A thread had an average of tensile

strength of 86.7 kg with a standard deviation of 6.28

kg, while type B had an average tensile strength of

77.8 with a standard deviation of 5.61 kg. Test the

manufacturer’s

claim

using

a

0.05

level

of

significance.


1
Expert's answer
2022-05-17T09:21:46-0400

Testing for Equality of Variances

A F-test is used to test for the equality of variances. The following F-ratio is obtained:


F=s12s22=6.2825.612=1.2531F=\dfrac{s_1^2}{s_2^2}=\dfrac{6.28^2}{5.61^2}=1.2531

The critical values for α=0.05,\alpha=0.05, df1=n11=49df_1=n_1-1=49 degrees of freedom, df2=n21=49df_2=n_2-1=49 degrees of freedom are FL=0.5675F_L = 0.5675 and FU=1.7622,F_U = 1.7622, and since F=1.2531F = 1.2531 then the null hypothesis of equal variances is not rejected.

The following null and alternative hypotheses need to be tested:

H0:μ1μ2H_0:\mu_1\le\mu_2

H1:μ1>μ2H_1:\mu_1>\mu_2

This corresponds to a right-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

The degrees of freedom are computed as follows, assuming that the population variances are equal:


df=df1+df2=n1+n22=98df=df_1+df_2=n_1+n_2-2=98

Based on the information provided, the significance level is α=0.05,\alpha = 0.05, and the degrees of freedom are df=98.df = 98.

The critical value for this right-tailed test, α=0.05,df=98\alpha = 0.05, df=98 degrees of freedom is tc=1.660551.t_c =1.660551.

The rejection region for this right-tailed test is R={t:t>1.660551}.R = \{t: t > 1.660551\}.

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:


t=Xˉ1Xˉ2(n11)s12+(n21)s22n1+n22(1n1+1n2)t=\dfrac{\bar{X}_1-\bar{X}_2}{\sqrt{\dfrac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}(\dfrac{1}{n_1}+\dfrac{1}{n_2})}}

=86.777.8(501)(6.28)2+(501)(5.61)250+502(150+150)=\dfrac{86.7-77.8}{\sqrt{\dfrac{(50-1)(6.28)^2+(50-1)(5.61)^2}{50+50-2}(\dfrac{1}{50}+\dfrac{1}{50})}}

=7.473426=7.473426

Since it is observed that t=7.473426>1.660551=tc,t = 7.473426>1.660551= t_c, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value for right-tailed, df=98df=98 degrees of freedom, t=7.473426t=7.473426 is p0,p\approx0, and since p=0<0.05=α,p=0<0.05=\alpha, it is concluded that the null hypothes is rejected.

Therefore, there is enough evidence to claim that the population mean μ1\mu_1 is greater than μ2,\mu_2, at the α=0.05\alpha = 0.05 significance level.


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