Respuesta :
Answer:
[tex]z=\frac{0.27 -0.31}{\sqrt{\frac{0.31(1-0.31)}{500}}}=-1.933[/tex]
[tex]p_v =P(Z<-1.933)=0.0266[/tex]
If we compare the p value obtained and the significance level given [tex]\alpha=0.02[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL reject the null hypothesis, and we can said that at 2% of significance the proportion of arrests that were not prosecuted is not significanlty less than 0.31.
Step-by-step explanation:
1) Data given and notation
n=500 represent the random sample taken
X represent the number of arrests that were not prosecuted.
[tex]\hat p=0.27[/tex] estimated proportion of arrests that were not prosecuted
[tex]p_o=0.31[/tex] is the value that we want to test
[tex]\alpha=0.02[/tex] represent the significance level
Confidence=98% or 0.98
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
2) Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that the true proportion is less than 0.31.:
Null hypothesis:[tex]p\geq 0.31[/tex]
Alternative hypothesis:[tex]p < 0.31[/tex]
When we conduct a proportion test we need to use the z statisitc, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
3) Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.27 -0.31}{\sqrt{\frac{0.31(1-0.31)}{500}}}=-1.933[/tex]
4) Statistical decision
It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.
The significance level provided [tex]\alpha=0.02[/tex]. The next step would be calculate the p value for this test.
Since is a left tailed test the p value would be:
[tex]p_v =P(Z<-1.933)=0.0266[/tex]
If we compare the p value obtained and the significance level given [tex]\alpha=0.02[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL reject the null hypothesis, and we can said that at 2% of significance the proportion of arrests that were not prosecuted is not significanlty less than 0.31.
