Please no Handwriting its hard to interpret -Thanks For this question, use a fai
ID: 3134125 • Letter: P
Question
Please no Handwriting its hard to interpret -Thanks
For this question, use a fair coin to take some samples and analyze them. First, take any fair coin and flip it 14 times. Count the number of heads out of the 14 flips. This is your first sample. Do this 4 more times and count the number of heads out of the 14 flips in each sample. Thus, you should have 5 samples of 14 flips each. The important number is the number of heads in each sample (this can be any whole number between 0 and 14). Then answer the following questions:
1-First, list the number of heads in each of the 5 samples.
2- Combining all 5 samples, calculate the mean number of heads. Show your work.
3- Combining all 5 samples, calculate the median number of heads. Show your work.
4- If you had a many samples, what value(s) should the mean and median have? Why?
5-For each of your 5 samples of 14 flips, use the binomial distribution to test the null hypothesis that the coin is fair, using = .05 (two-tailed). (The alternative hypothesis is that the coin is not fair.) For each sample, calculate the p value (the probability of getting the number of heads you did or a more extreme number of heads (“two-tailed”) purely by chance). Show your work and organize the data into a small table. For each sample indicate the number of heads out of 14, the p value, your decision about the null hypothesis, and the explicit reason for your decision, in terms of the decision rule.
Explanation / Answer
For this question, use a fair coin to take some samples and analyze them. First, take any fair coin and flip it 14 times. Count the number of heads out of the 14 flips. This is your first sample. Do this 4 more times and count the number of heads out of the 14 flips in each sample. Thus, you should have 5 samples of 14 flips each. The important number is the number of heads in each sample (this can be any whole number between 0 and 14). Then answer the following questions:
Solution:
The five samples for flipping a coin 14 times are given as below:
0 = Tail
1 = Head
First sample
Second sample
Third sample
Fourth sample
Fifth sample
1
1
1
0
0
0
0
0
1
0
0
1
0
1
1
1
1
0
1
0
1
1
1
0
1
0
0
1
1
0
1
1
0
0
0
1
0
0
1
1
1
0
1
0
0
1
0
0
0
1
0
1
0
1
0
0
0
0
1
1
0
0
0
1
1
1
1
1
0
1
Number of Heads
8
7
5
8
7
1-First, list the number of heads in each of the 5 samples.
Solution:
Number of heads in each of the 5 samples is given as below:
Sample
First sample
Second sample
Third sample
Fourth sample
Fifth sample
Number of Heads
8
7
5
8
7
2- Combining all 5 samples, calculate the mean number of heads. Show your work.
Solution:
We get, total number of heads in all five samples = 35
Number of samples = 5
Mean = 35/5 = 7
Mean = 7
3- Combining all 5 samples, calculate the median number of heads. Show your work.
Solution:
Median = middle most observation in observations in increasing order
Observations in increasing order are given as below:
Median
Number of Heads
5
7
7
8
8
Middle Most Observation
Median = 7
4- If you had a many samples, what value(s) should the mean and median have? Why?
Solution:
If we had a many samples, then we should have the mean = 7 and median = 7 because this is unbiased estimators for mean and median as the number of samples increases.
5-For each of your 5 samples of 14 flips, use the binomial distribution to test the null hypothesis that the coin is fair, using = .05 (two-tailed). (The alternative hypothesis is that the coin is not fair.) For each sample, calculate the p value (the probability of getting the number of heads you did or a more extreme number of heads (“two-tailed”) purely by chance). Show your work and organize the data into a small table. For each sample indicate the number of heads out of 14, the p value, your decision about the null hypothesis, and the explicit reason for your decision, in terms of the decision rule.
Solution:
Here, we have to check the claim whether the coin is fair or not. For checking this claim, we have to assume that the proportion of heads should be 0.5 or 50% if the coin is fair. Here, we have to use the one sample z test for the population proportion. The null and alternative hypothesis for this test is given as below:
Null hypothesis: H0: The coin is fair or population proportion for number of heads is 0.5.
Alternative hypothesis: Ha: The coin is not fair or population proportion for number of heads is not 0.5.
Symbolically, it is given as below:
H0: P = 0.5 versus Ha: P 0
This is a two tailed test.
We have given that the level of significance = alpha = 0.05 or 5%
We have N = 70
Number of heads = 35
The test is given as below:
Z Test of Hypothesis for the Proportion
Data
Null Hypothesis p =
0.5
Level of Significance
0.05
Number of Items of Interest
35
Sample Size
70
Intermediate Calculations
Sample Proportion
0.5
Standard Error
0.0598
Z Test Statistic
0.0000
Two-Tail Test
Lower Critical Value
-1.9600
Upper Critical Value
1.9600
p-Value
1.0000
Do not reject the null hypothesis
Here, we get the p-value greater than the given level of significance, so we do not reject the null hypothesis that the coin is fair or population proportion for number of heads is 0.5.
First sample
Second sample
Third sample
Fourth sample
Fifth sample
1
1
1
0
0
0
0
0
1
0
0
1
0
1
1
1
1
0
1
0
1
1
1
0
1
0
0
1
1
0
1
1
0
0
0
1
0
0
1
1
1
0
1
0
0
1
0
0
0
1
0
1
0
1
0
0
0
0
1
1
0
0
0
1
1
1
1
1
0
1
Number of Heads
8
7
5
8
7
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.