Question

Please do as little intermediate rounding as possible in order to get the correct p-value.
In a sample of 200 of Deadpool's severed arms, 112 of them grow back a new Deadpool. In a sample of 210 of Deadpool's legs, 90 of them grow back a new Deadpool.  
Do we have evidence at the various levels that Deadpool's severed arms are more likely to grow back a new Deadpool than his severed legs?
The associated p-value for this hypothesis test is ( )? (Answers to four places after the decimal.)

(a) At the 13% level (yes or no?)

(b)At the 10% level (yes or no?)

(c)At the 7% level (yes or no?)

(d)At the 5% level (yes or no?)

(e)At the 3% level (yes or no?)

(f)At the 2% level (yes or no?)

(g)At the 1% level (yes or no?)

(h)At the 0.2% level (yes or no?)

(i)At the 0.1% level (yes or no?)

Answer 1

From the question, we have the following.

Sample 1: In a sample of 200 of Deadpool's severed arms, 112 of them grow back a new Deadpool.

Sample 2: In a sample of 210 of Deadpool's legs, 90 of them grow back a new Deadpool.

We have to check whether Deadpool's severed arms are more likely to grow back a new Deadpool than his severed legs.

Here let us consider the grow back rate as a population characteristic. Then Sample 1 has 112 of 200 which posses this characteristic and similarly Sample 2 has 90 out of 210 which has this characteristic.

In this case, the question can be now restated equivalently as follows:

Using the information from the samples, we have to check whether the proportion of the population which posses this characteristic is the same in both population. i.e., we have to check for equality of proportions.

Let P1 and P2 denote the population proportion which posses the characteristic of grow back, for the populations of Deadpool's severed arms and legs respectively.

Then the hypothesis of the test can be stated as to test:

grow back rate is the same for Deadpool's severed arms and legs.

Deadpool's severed arms are more likely to grow back a new Deadpool than his severed legs.

i.e.,

This test can be easily performed using R-software. Please find attached the screenshots with codes and the corresponding results.

1. Data preparation
   Following are the scripts for data preparation.
   
   
   The R-output of the prepared data looks as below.
   
   
   
2. Testing the hyposthesis

   The associated p-value for this hypothesis test is ( )? (Answers to four places after the decimal.)
   Following are the scripts for testing.
   
   
   The R-output of the of the above test is given below.
   
   From the output it is clear that the p - value for this test is 0.0039.
   
   The answers for all subpart questions can be reached as follows:
   For all the tests where the p-value (= 0.0039) is less than the given significance level, we fail to accept the null hypothesis. Alternatively, for all the tests where the p-value (= 0.0039) is greater than the given significance level, we fail to reject the null hypothesis.
   i.e., we fail to accept the null hypothesis for parts (a) to (g). and we fail to reject the null hypothesis for parts (h) and (g).
   
   
   
   
   Please find the detailed explanation for the subparts below.

   (a) At the 13% level (yes or no?)
   Here, we have to conduct the test at a significance level of 13%.
   Therefore, we have to conduct the test a confidence level of 100 - 13 = 87.
   
   Following are the scripts for this test.
   
   The R-output of the of the above test is given below.
   
   Here, since the p-value (= 0.0039) is less than the significance level (= 0.13), we fail to accept the null hypothesis.
   This is favorable to the conclusion that the Deadpool's severed arms are more likely to grow back a new Deadpool than his severed legs.

   (b)At the 10% level (yes or no?)
   Here, we have to conduct the test at a significance level of 10%.
   Therefore, we have to conduct the test a confidence level of 100 - 10 = 90.
   
   Following are the scripts for this test.
   
   The R-output of the of the above test is given below.
   
   Here also, since the p-value (= 0.0039) is less than the significance level (= 0.10), we fail to accept the null hypothesis.
   This is favorable to the conclusion that the Deadpool's severed arms are more likely to grow back a new Deadpool than his severed legs.

   (c)At the 7% level (yes or no?)
   Here, we have to conduct the test at a significance level of 7%.
   Therefore, we have to conduct the test a confidence level of 100 - 7 = 93.
   
   Following are the scripts for this test.
   
   The R-output of the of the above test is given below.
   
   Here also, since the p-value (= 0.0039) is less than the significance level (= 0.07), we fail to accept the null hypothesis.
   This is favorable to the conclusion that the Deadpool's severed arms are more likely to grow back a new Deadpool than his severed legs.

   (d)At the 5% level (yes or no?)
   Here, we have to conduct the test at a significance level of 5%.
   Therefore, we have to conduct the test a confidence level of 100 - 5 = 95.
   
   Following are the scripts for this test.
   
   The R-output of the of the above test is given below.
   
   Here also, since the p-value (= 0.0039) is less than the significance level (= 0.05), we fail to accept the null hypothesis.
   This is favorable to the conclusion that the Deadpool's severed arms are more likely to grow back a new Deadpool than his severed legs.

   (e)At the 3% level (yes or no?)
   Here, we have to conduct the test at a significance level of 3%.
   Therefore, we have to conduct the test a confidence level of 100 - 3 = 97.
   
   Following are the scripts for this test.
   
   The R-output of the of the above test is given below.
   
   Here also, since the p-value (= 0.0039) is less than the significance level (= 0.03), we fail to accept the null hypothesis.
   This is favorable to the conclusion that the Deadpool's severed arms are more likely to grow back a new Deadpool than his severed legs.

   (f)At the 2% level (yes or no?)
   Here, we have to conduct the test at a significance level of 2%.
   Therefore, we have to conduct the test a confidence level of 100 - 2 = 98.
   
   Following are the scripts for this test.
   
   The R-output of the of the above test is given below.
   
   Here also, since the p-value (= 0.0039) is less than the significance level (= 0.02), we fail to accept the null hypothesis.
   This is favorable to the conclusion that the Deadpool's severed arms are more likely to grow back a new Deadpool than his severed legs.

   (g)At the 1% level (yes or no?)
   Here, we have to conduct the test at a significance level of 1%.
   Therefore, we have to conduct the test a confidence level of 100 - 1 = 99.
   
   Following are the scripts for this test.
   
   The R-output of the of the above test is given below.
   
   Here also, since the p-value (= 0.0039) is less than the significance level (= 0.01), we fail to accept the null hypothesis.
   This is favorable to the conclusion that the Deadpool's severed arms are more likely to grow back a new Deadpool than his severed legs.

   (h)At the 0.2% level (yes or no?)
   Here, we have to conduct the test at a significance level of 0.2%.
   Therefore, we have to conduct the test a confidence level of 100 - 0.2 = 99.8
   
   Following are the scripts for this test.
   
   The R-output of the of the above test is given below.
   
   Here, since the p-value (= 0.0039) is greater than the significance level (= 0.002), we fail to reject the null hypothesis.
   This is favorable to the conclusion that the grow back rate is the same for Deadpool's severed arms and legs..

   (i)At the 0.1% level (yes or no?)

   Here, we have to conduct the test at a significance level of 0.1%.
   Therefore, we have to conduct the test a confidence level of 100 - 0.1 = 99.9
   
   Following are the scripts for this test.
   
   The R-output of the of the above test is given below.
   
   Here, since the p-value (= 0.0039) is greater than the significance level (= 0.001), we fail to reject the null hypothesis.
   This is favorable to the conclusion that the grow back rate is the same for Deadpool's severed arms and legs..