Using the t-distribution, it is found that:
a) Since the p-value of the test is 0.15 > 0.05, there is not significant evidence to conclude that the mean weight is below 325-mg, that is, that it meets specifications.
b) Any larger sample size does not ensure that regulations are met, it has to be significantly large such that t will be less than the critical value for the test.
At the null hypothesis, we<u> test if the mean sodium content is of at least 325-mg</u>, that is:
![H_0: \mu \geq 325](https://tex.z-dn.net/?f=H_0%3A%20%5Cmu%20%5Cgeq%20325)
At the alternative hypothesis, we <u>test if it is less than 325-mg</u>, that is:
![H_1: \mu < 325](https://tex.z-dn.net/?f=H_1%3A%20%5Cmu%20%3C%20325)
Item a:
We have the <u>standard deviation for the sample</u>, thus, the t-distribution is used. The test statistic is given by:
The parameters are:
is the sample mean.
is the value tested at the null hypothesis.
- s is the standard deviation of the sample.
- n is the sample size.
For this problem, the values of the <u>parameters</u> are: ![\overline{x} = 322, \mu = 325, s = 18, n = 40](https://tex.z-dn.net/?f=%5Coverline%7Bx%7D%20%3D%20322%2C%20%5Cmu%20%3D%20325%2C%20s%20%3D%2018%2C%20n%20%3D%2040)
The value of the <u>test statistic</u> is:
![t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7B%5Coverline%7Bx%7D%20-%20%5Cmu%7D%7B%5Cfrac%7Bs%7D%7B%5Csqrt%7Bn%7D%7D%7D)
![t = \frac{322 - 325}{\frac{18}{\sqrt{40}}}](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7B322%20-%20325%7D%7B%5Cfrac%7B18%7D%7B%5Csqrt%7B40%7D%7D%7D)
![t = -1.05](https://tex.z-dn.net/?f=t%20%3D%20-1.05)
The p-value of the test is found using a left-tailed test, as we test if the mean is less than a value, with <u>t = -1.05</u> and <u>40 - 1 = 39 df</u>.
Using a t-distribution calculator, this p-value is of 0.15.
Since the p-value of the test is 0.15 > 0.05, there is not significant evidence to conclude that the mean weight is below 325-mg, that is, that it meets specifications.
Item b:
- Increasing the sample size decreases t which also decreases the p-value in a left-tailed test.
- However, the specifications are only met is t is lower than the critical value of t, hence, there has to be a significant increase in sample size.
Hence:
Any larger sample size does not ensure that regulations are met, it has to be significantly large such that t will be less than the critical value for the test.
A similar problem is given at brainly.com/question/25454581