Answer:
The z-test statistic for this hypothesis test is ![z = 4.33](https://tex.z-dn.net/?f=z%20%3D%204.33)
Step-by-step explanation:
Proportion in 2000:
10 of the 50 men were obese, so:
![p = \frac{10}{50} = 0.2](https://tex.z-dn.net/?f=p%20%3D%20%5Cfrac%7B10%7D%7B50%7D%20%3D%200.2)
Test if it has increased:
At the null hypothesis, we test if the prevalence of obesity has not increased, that is, the proportion is of 0.2 or less, so:
![H_0: p \leq 0.2](https://tex.z-dn.net/?f=H_0%3A%20p%20%5Cleq%200.2)
At the alternative hypothesis, we test if this prevalence has increased, that is, the proportion is above 0.2. So
![H_1: p > 0.2](https://tex.z-dn.net/?f=H_1%3A%20p%20%3E%200.2)
The test statistic is:
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.
0.2 is tested at the null hypothesis:
This means that ![\mu = 0.2, \sigma = \sqrt{0.2(1-0.2)} = 0.4](https://tex.z-dn.net/?f=%5Cmu%20%3D%200.2%2C%20%5Csigma%20%3D%20%5Csqrt%7B0.2%281-0.2%29%7D%20%3D%200.4)
30 out of the 75 men from 2010 were assigned as obese.
This means that ![n = 75, X = \frac{30}{75} = 0.4](https://tex.z-dn.net/?f=n%20%3D%2075%2C%20X%20%3D%20%5Cfrac%7B30%7D%7B75%7D%20%3D%200.4)
Value of the z-test statistic:
![z = 4.33](https://tex.z-dn.net/?f=z%20%3D%204.33)
The z-test statistic for this hypothesis test is ![z = 4.33](https://tex.z-dn.net/?f=z%20%3D%204.33)