Respuesta :
Answer:
a1) The null and alternative hypothesis are:
[tex]H_0: \pi=0.5\\\\H_a:\pi>0.5[/tex]
Reject H0 if the test statistic is larger than 1.282 (z>1.282)
a2) Test statistic z=1.94
a3) True.
a4) True.
b1) P-value=0.0264
b2) Yes, it is biased towards heads.
Step-by-step explanation:
This is a hypothesis test for a proportion.
The claim is that the coin is biased towards heads.
Then, the null and alternative hypothesis are:
[tex]H_0: \pi=0.5\\\\H_a:\pi>0.5[/tex]
The significance level is 0.10.
The sample has a size n=60.
The sample proportion is p=0.6333.
[tex]p=X/n=38/60=0.6333[/tex]
The standard error of the proportion is:
[tex]\sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt{\dfrac{0.5*0.5}{60}}\\\\\\ \sigma_p=\sqrt{0.00417}=0.0645[/tex]
Then, we can calculate the z-statistic as:
[tex]z=\dfrac{p-\pi+0.5/n}{\sigma_p}=\dfrac{0.6333-0.5-0.5/60}{0.0645}=\dfrac{0.125}{0.0645}=1.936[/tex]
This test is a right-tailed test, so the P-value for this test is calculated as:
[tex]P-value=P(z>1.936)=0.0264[/tex]
As the P-value (0.0264) is smaller than the significance level (0.10), the effect is significant.
The null hypothesis is rejected.
There is enough evidence to support the claim that the coin is biased towards heads.
The critical value for a significance level of 0.1 is z=1.282. As this is a right-tailed test, the decision rule is: "Reject H0 if the test statistic is larger than 1.282 (z>1.282)", as the rejection region is for every z over 1.282.
