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
and
the conclusion is that the thread has become inferior
Step-by-step explanation:
From the question we are told that
The population mean is ![\mu = 9.72 \ oz](https://tex.z-dn.net/?f=%5Cmu%20%3D%20%209.72%20%5C%20oz)
The standard deviation is ![\sigma = 1.40\ oz](https://tex.z-dn.net/?f=%5Csigma%20%20%3D%201.40%5C%20%20oz)
The sample size is n = 36
The sample mean is ![\= x = 8.93 \ oz](https://tex.z-dn.net/?f=%5C%3D%20x%20%20%3D%20%208.93%20%5C%20oz)
The null hypothesis is ![H_o : \mu = 9.72 \ oz](https://tex.z-dn.net/?f=H_o%20%20%3A%20%20%5Cmu%20%3D%20%209.72%20%5C%20%20oz)
The alternative hypothesis is ![H_a : \mu < 9.72 \ oz](https://tex.z-dn.net/?f=H_a%20%20%3A%20%20%5Cmu%20%3C%20%209.72%20%5C%20oz)
Generally the test statistics is mathematically represented as
![t = \frac{ \= x - \mu }{ \frac{\sigma }{ \sqrt{n} } }](https://tex.z-dn.net/?f=t%20%3D%20%20%5Cfrac%7B%20%5C%3D%20x%20-%20%5Cmu%20%7D%7B%20%5Cfrac%7B%5Csigma%20%7D%7B%20%5Csqrt%7Bn%7D%20%7D%20%7D)
=> ![t = \frac{ 8.93 -9.72}{ \frac{ 1.4 }{ \sqrt{36} } }](https://tex.z-dn.net/?f=t%20%3D%20%20%5Cfrac%7B%208.93%20-9.72%7D%7B%20%5Cfrac%7B%201.4%20%7D%7B%20%5Csqrt%7B36%7D%20%7D%20%7D)
=> ![t = -3.33](https://tex.z-dn.net/?f=t%20%3D%20-3.33)
So
The p-value obtained from the z- table is
![p-value = P( Z < -3.39) = 0.00034946](https://tex.z-dn.net/?f=p-value%20%20%3D%20%20P%28%20Z%20%20%3C%20%20-3.39%29%20%3D%200.00034946)
So at ![\alpha = 0.0 5](https://tex.z-dn.net/?f=%5Calpha%20%20%3D%20%200.0%205)
![p-value < \alpha](https://tex.z-dn.net/?f=p-value%20%20%3C%20%20%5Calpha)
So we reject the null hypothesis,hence we conclude that the thread has become inferior
So at ![\alpha = 0.0 1](https://tex.z-dn.net/?f=%5Calpha%20%20%3D%20%200.0%201)
![p-value < \alpha](https://tex.z-dn.net/?f=p-value%20%20%3C%20%20%5Calpha)
So we reject the null hypothesis,hence we conclude that the thread has become inferior