Complete Question
It has been found from experience that the mean breaking strength of a particular brand of thread is 9.72 oz with a standard deviation of 1.4 oz. Recently a sample of 36 pieces of thread showed a mean breaking strength of 8.93 oz. Can one conclude at a significance level of (a) 0.05, (b) 0.01 that the thread has become inferior?
Answer:
At both [tex]\alpha = 0.05[/tex] and [tex]\alpha = 0.01[/tex] the conclusion is that the thread has become inferior
Step-by-step explanation:
From the question we are told that
The population mean is [tex]\mu = 9.72 \ oz[/tex]
The standard deviation is [tex]\sigma = 1.40\ oz[/tex]
The sample size is n = 36
The sample mean is [tex]\= x = 8.93 \ oz[/tex]
The null hypothesis is [tex]H_o : \mu = 9.72 \ oz[/tex]
The alternative hypothesis is [tex]H_a : \mu < 9.72 \ oz[/tex]
Generally the test statistics is mathematically represented as
[tex]t = \frac{ \= x - \mu }{ \frac{\sigma }{ \sqrt{n} } }[/tex]
=> [tex]t = \frac{ 8.93 -9.72}{ \frac{ 1.4 }{ \sqrt{36} } }[/tex]
=> [tex]t = -3.33[/tex]
So
The p-value obtained from the z- table is
[tex]p-value = P( Z < -3.39) = 0.00034946[/tex]
So at [tex]\alpha = 0.0 5[/tex]
[tex]p-value < \alpha[/tex]
So we reject the null hypothesis,hence we conclude that the thread has become inferior
So at [tex]\alpha = 0.0 1[/tex]
[tex]p-value < \alpha[/tex]
So we reject the null hypothesis,hence we conclude that the thread has become inferior
