Respuesta :
Answer:
The calculated value of test statistic is z=2.48.
This has a P-value of P=0.00657.
If we state the null hypothesis [tex]H_0: \pi\leq0.44[/tex] at a significance level of [tex]\alpha=0.05[/tex], we would reject this null hypothesis as [tex]P-value<\alpha[/tex].
Step-by-step explanation:
We have in this problem, a hypothesis test of proportions.
The test statistic for this is the z-value, and is calculated like that:
[tex]z=\frac{p-\pi-0.5/N}{\sigma}[/tex]
Where the term 0.5/N is the correction for continuity and is negative in the cases that p>π.
p: proportion of the sample; π: proportion of the population; σ: standard deviation of the population.
The standard deviation of the population has to be calculated as:
[tex]\sigma=\sqrt{\frac{\pi(1-\pi)}{N} } =\sqrt{\frac{0.44(1-0.44)}{250} }=\sqrt{0.0009856}=0.0314[/tex]
The proportion of the sample (p) is [tex]p=130/250=0.52[/tex].
Then, the test statistic z is
[tex]z=\frac{p-\pi-0.5/N}{\sigma}=\frac{0.52-0.44-0.5/250}{0.0314} =\frac{0.078}{0.0314} =2.48[/tex]
The P-value of this statistic is P(z>2.48)=0.00657
If we state the null hypothesis [tex]H_0: \pi\leq0.44[/tex] at a significance level of [tex]\alpha=0.05[/tex], we would reject this null hypothesis as [tex]P-value<\alpha[/tex].
