Answer:
0 < 0.05, which means that we reject the null hypothesis, meaning that the air pressure of the balls is different of the target value of 7.9.
Step-by-step explanation:
The air pressure of a particular ball has a target value of 7.9 PSI.
This means that the null hypothesis is:
![H_{0}: \mu = 7.9](https://tex.z-dn.net/?f=H_%7B0%7D%3A%20%5Cmu%20%3D%207.9)
The alternate hypothesis is:
![H_{a}: \mu \neq 7.9](https://tex.z-dn.net/?f=H_%7Ba%7D%3A%20%5Cmu%20%5Cneq%207.9)
The test statistic is:
![z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D)
In which X is the sample mean,
is the value tested at the null hypothesis,
is the standard deviation and n is the size of the sample.
The hypothesis tested means that ![\mu = 7.9](https://tex.z-dn.net/?f=%5Cmu%20%3D%207.9)
Suppose the basketballs have a normal distribution with a standard deviation of 0.20 PSI.
This means that ![\sigma = 0.2](https://tex.z-dn.net/?f=%5Csigma%20%3D%200.2)
When a shipment of basketballs arrive, the consumer takes a sample of 21 from the shipment and measures their PSI to see if it meets the target value, and finds the mean to be 7.3 PSI.
This means that ![X = 7.3, n = 21](https://tex.z-dn.net/?f=X%20%3D%207.3%2C%20n%20%3D%2021)
The test statistic is:
![z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D)
![z = \frac{7.3 - 7.9}{\frac{0.2}{\sqrt{21}}}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B7.3%20-%207.9%7D%7B%5Cfrac%7B0.2%7D%7B%5Csqrt%7B21%7D%7D%7D)
![z = -13.74](https://tex.z-dn.net/?f=z%20%3D%20-13.74)
pvalue:
We are testing that the mean pressure is different than the target value of 7.9, and since the test statistic is negative, the pvalue is 2 multiplied by the pvalue of z = -13.74, which we find looking at the z-table.
has a pvalue of 0.
2*0 = 0
0 < 0.05, which means that we reject the null hypothesis, meaning that the air pressure of the balls is different of the target value of 7.9.