Answer:
From the question we are told that
The population mean is
The sample size is n = 30
The sample mean is ![\= x = 16.32](https://tex.z-dn.net/?f=%5C%3D%20x%20%3D%20%2016.32)
The population standard deviation is ![\sigma = 0.8](https://tex.z-dn.net/?f=%5Csigma%20%20%3D%20%200.8)
The level of significance is ![\alpha = 0.10](https://tex.z-dn.net/?f=%5Calpha%20%20%3D%200.10)
Step 1: State hypotheses:
The null hypothesis is ![H_o : \mu = 16](https://tex.z-dn.net/?f=H_o%20%3A%20%20%5Cmu%20%3D%2016)
The alternative hypothesis is ![H_a : \mu \ne 16](https://tex.z-dn.net/?f=H_a%20%3A%20%20%5Cmu%20%5Cne%20%2016)
Step 2: State the test statistic. Since we know the population standard deviation and the sample is large our test statistics is
![t = \frac{ \= x -\mu }{ \frac{\sigma}{ \sqrt{n} } }](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7B%20%5C%3D%20x%20%20-%5Cmu%20%7D%7B%20%5Cfrac%7B%5Csigma%7D%7B%20%5Csqrt%7Bn%7D%20%7D%20%7D)
=> ![t = \frac{ 16.32 -16 }{ \frac{0.8 }{ \sqrt{30} } }](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7B%2016.32%20%20-16%20%20%7D%7B%20%5Cfrac%7B0.8%20%7D%7B%20%5Csqrt%7B30%7D%20%7D%20%7D)
=> ![t =2.191](https://tex.z-dn.net/?f=t%20%3D2.191)
Generally the degree of freedom is mathematically represented as
![df = n - 1](https://tex.z-dn.net/?f=df%20%20%3D%20%20n%20-%201)
=> ![df = 30 - 1](https://tex.z-dn.net/?f=df%20%20%3D%20%2030%20%20-%201)
=> ![df =29](https://tex.z-dn.net/?f=df%20%20%3D29)
Step 3: State the critical region(s):
From the student t-distribution table the critical value corresponding to
is
![t = 1.311](https://tex.z-dn.net/?f=t%20%3D%201.311)
Generally the critical regions is mathematically represented as
![- 1.311 < T < 1.311](https://tex.z-dn.net/?f=-%201.311%20%3C%20T%20%3C%20%201.311)
Step 4: Conduct the experiment/study:
Generally the from the value obtained we see that the t value is outside the critical region so the decision is [Reject the null hypothesis ]
Step 5: Reach conclusions and state in English:
There is sufficient evidence to show that the filling weight has to be adjusted
Step 6: Calculate the p-value associated with this test. How does this the p-value support your conclusions in Step 5?
From the student t-distribution table the probability value to the right corresponding to
at a degree of freedom of
is
![P( t > 2.191) = 0.0183](https://tex.z-dn.net/?f=P%28%20t%20%3E%202.191%29%20%3D%20%200.0183)
Generally the p-value is mathematically represented as
![p-value = 2 * P( t > 2.191 )](https://tex.z-dn.net/?f=p-value%20%20%3D%202%20%2A%20P%28%20t%20%3E%20%202.191%20%29)
=> ![p-value = 2 * 0.0183](https://tex.z-dn.net/?f=p-value%20%20%3D%202%20%2A%200.0183)
=> ![p-value = 0.0366](https://tex.z-dn.net/?f=p-value%20%20%3D%20%200.0366)
Generally looking at the value obtained we see that
hence
The decision rule is
Reject the null hypothesis
Step-by-step explanation: