Answer:
a) H₀ : p = 0.5
![H_{a} : p > 0.5](https://tex.z-dn.net/?f=H_%7Ba%7D%20%3A%20p%20%3E%200.5)
b) z = 2.0785
c) p - value = 0.0188
d) Our conclusion is that there has been a significant increase in the proportion of registered voters who are planning to vote for the incumbent president
Explanation:
Probability of those planning to vote for the incumbent president before the debate, P₀ = 50/100 = 0.5
a) The null and the alternative hypothesis
Let the null hypothesis be represented by ![H_{0}](https://tex.z-dn.net/?f=H_%7B0%7D)
and the alternative hypothesis be represented by ![H_{a}](https://tex.z-dn.net/?f=H_%7Ba%7D)
H₀ : p = 0.5
![H_{a} : p > 0.5](https://tex.z-dn.net/?f=H_%7Ba%7D%20%3A%20p%20%3E%200.5)
b) compute the test statistics
sample size, n = 1200
number of voters planning to vote for the incumbent president, x = 636
Probability of those planning to vote for the incumbent president in the selected sample:
![\bar{p} = x/n\\\bar{p} = 636/1200\\\bar{p} = 0.53](https://tex.z-dn.net/?f=%5Cbar%7Bp%7D%20%3D%20x%2Fn%5C%5C%5Cbar%7Bp%7D%20%3D%20636%2F1200%5C%5C%5Cbar%7Bp%7D%20%3D%200.53)
Formula for the test statistic:
![z = \frac{\bar{p} - p_{0} }{\sqrt{\frac{p_{0}(1-p_{0} ) }{n} } }](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B%5Cbar%7Bp%7D%20-%20p_%7B0%7D%20%7D%7B%5Csqrt%7B%5Cfrac%7Bp_%7B0%7D%281-p_%7B0%7D%20%29%20%7D%7Bn%7D%20%7D%20%7D)
![z = \frac{0.53 - 0.5 }{\sqrt{\frac{0.5(1-0.5 ) }{1200} } }](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B0.53%20-%200.5%20%7D%7B%5Csqrt%7B%5Cfrac%7B0.5%281-0.5%20%29%20%7D%7B1200%7D%20%7D%20%7D)
z = 2.0785
c)
For
, the critical value as checked from the normal table will be:
![z_{\alpha} = 1.645](https://tex.z-dn.net/?f=z_%7B%5Calpha%7D%20%3D%201.645)
The P value for z > 2.0785
![P-value = P(z > 2.0785) \\P - value = \phi (-2.0785)\\P - value = 0.0188](https://tex.z-dn.net/?f=P-value%20%3D%20P%28z%20%3E%202.0785%29%20%5C%5CP%20-%20value%20%3D%20%5Cphi%20%28-2.0785%29%5C%5CP%20-%20value%20%3D%200.0188)
d) What do you conclude?
since
, we will reject the null hypothesis.
Our conclusion is that there has been a significant increase in the proportion of registered voters who are planning to vote for the incumbent president