Question #194451

A manufacture of rayon wants to compare that the yield strength of 5 11⋅ kg/ 2 mm is 

met or not at 5% level of significance. The manufacturer draws a sample and 

calculates the mean to be 8 12⋅ kg/ 2 mm and the standard derivation is known to be 

2⋅0kg/ mm .

2

 Carry out the statistical test appropriate for this.


1
Expert's answer
2021-05-19T07:57:46-0400

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H0:μ=5.11 kg/mm2H_0: \mu=5.11\ kg/mm^2

H1:μ5.11 kg/mm2H_1: \mu\not=5.11\ kg/mm^2

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is α=0.05\alpha=0.05 and df=n1=301=29df=n-1=30-1=29 degrees of freedom.

The critical value for a two-tailed test is tc=2.04523.t_c=2.04523.  

The rejection region for this two-tailed test is R={t:t>2.04523}.R=\{t:|t|>2.04523\}.


The tt - statistic is computed as follows:


t=xˉμs/n=8.125.112.0/308.24322t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{8.12-5.11}{2.0/\sqrt{30}}\approx8.24322

Since it is observed that t=8.24322>2.04523=tc,|t|=8.24322>2.04523=|t_c|, it is then concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean μ\mu  is different than 5.11,5.11, at the α=0.05\alpha=0.05 significance level.


Using the P-value approach: The p-value for two-tailed, the significance level  α=0.05,t=8.24322\alpha=0.05, t=8.24322 and df=29df=29 degrees of freedom is p<0.00001,p<0.00001,

and since p<0.00001<0.05=α,p<0.00001<0.05=\alpha, it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean μ\mu  is different than 5.11,5.11, at the α=0.05\alpha=0.05 significance level.


Therefore, there is enough evidence to claim that the yield strength is different than 5.11,5.11, at the α=0.05\alpha=0.05 significance level.



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