Respuesta :
Answer:
a) P-value=0.16
b) critic value zc=-1.645
c) area of the critical region = 0.05
Step-by-step explanation:
We have a hypothesis test on the population proportion.
The sample proportion is p=66/144=11/24=0.4583.
The z-statistic results in z=-1.
As the alternative hypothesis (the claim) states that the proportion is below 0.5, the test is left-tailed.
The P-value for this type of test and with a z=-1 is calculated as:
[tex]P-value=P(z<-1)=0.16[/tex]
The critial value depends on the significance level. In this test we have a significance level of 0.05.
The critical value can be looked-up in a standard normal distribution table. Its the value for which the probability of having a statistic below this critical value is equal to the significance level (0.05):
[tex]P(z<z_c)=0.05[/tex]
For this test, the critical value is zc=-1.645.
As the área of the critical region is equal to the probability of having a statistic below this critical value, and this is the significance level, we have a área of the critical region equal to 0.05.
Answer:
a) [tex]z=\frac{0.458 -0.5}{\sqrt{\frac{0.5(1-0.5)}{144}}}=-1.008[/tex]
Since is a left tailed test the p value would be:
[tex]p_v =P(z<-1.008)=0.1567[/tex]
b) [tex] z_{crit}= -1.645[/tex]
c) For this case since is a left tailed test the critical region or the rejection zone of the null hypothesis would be:
[tex] (\infty , -1.645)[/tex]
Step-by-step explanation:
Data given and notation
n=144 represent the random sample taken
X=66 represent the number of girls
[tex]\hat p=\frac{66}{144}=0.458[/tex] estimated proportion of girls
[tex]p_o=0.5[/tex] is the value that we want to test
[tex]\alpha=0.05[/tex] represent the significance level
Confidence=95% or 0.95
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that true proportion is less than 0.5.:
Null hypothesis:[tex]p\geq 0.5[/tex]
Alternative hypothesis:[tex]p < 0.5[/tex]
When we conduct a proportion test we need to use the z statistic, 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].
Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.458 -0.5}{\sqrt{\frac{0.5(1-0.5)}{144}}}=-1.008[/tex]
Part a : p value
The significance level provided [tex]\alpha=0.05[/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<-1.008)=0.1567[/tex]
Part b
We want to conduct a left tailed test with [tex]\alpha=0.05[/tex] and we need to find a critical value in the normal standard distribution who accumulates 0.05 of the area in the left and we got:
[tex] z_{crit}= -1.645[/tex]
Part c
For this case since is a left tailed test the critical region or the rejection zone of the null hypothesis would be:
[tex] (\infty , -1.645)[/tex]
