Introduction
In phase 3 of this stats project, we will continue to utilize our two samples of Flagler students. Our two samples, as previously defined, are taken from two survey questions that ask if both parents are still married and if the person being surveyed believes that it is important to keep in contact with both parents. In hypothesis test #1, the null hypothesis that 50% of all Flagler College students think that ongoing contact with both parent is important is tested. Then, the percentage of all Flagler students who believe contact with both parents is important is calculated and given a confidence interval to correspond with the level of certainty that may be attributed to that percentage. In hypothesis test #2, the null hypothesis that there is no difference in the responses given from people whose parents either stayed married or divorced as to whether it is important to maintain contact with both parents is tested. The percentage of both those whose parents are still married and those whose parents have divorced are then calculated in relation to their answers to the question of if it is important to maintain contact with both parents and assigned a confidence interval regarding the certainty that those percentages are accurate.
Hypothesis Test #1  Making a Majority Claim
150 students were sampled for the following question: “Do you think it is important for children of divorced parents to continue to have ongoing contact with both parents?” 123 students, or 82%, replied “Yes,” that they believed children should have ongoing contact. 82% is a majority in the sample, so we will be using the sample results to test the claim that in the population of Flagler College students, the majority believe children should have ongoing contact with both divorced parents. Our significance level will be set to 0.05. We will go through the four steps of the process for testing a hypothesis. A pie chart of the data is provided below.
Hypothesize
Null: 50% of all Flagler College students believe that children of divorced parents should have ongoing contact with both parents.
Alternate: More than 50% of all Flagler College students believe that children of divorced parents should have ongoing contact with both parents.
This is a rightsided test, as the alternative hypothesis indicates that we are only looking if the data indicates a “greater than” result.
Prepare
1. Random Sample – Though we are not certain if the sample is truly random, we will make the assumption that it is in order to conduct our test.
2. Large Sample – To check if the sample is large enough, we compute based on the null hypothesis and check if the numbers for both expected successes and failures are larger than ten.
np0= (150) (0.50) = 75 > 10
n(1p0) = (150) (0.50) = 75 > 10
Since both expected successes and failures are more than ten, the sample is large enough.
3. Big Population – Flagler College’s population is around 2500 students. For this condition to pass, the population must be more than 10x bigger than the sample.
10n = (10)(150) = 1500 < 2500
Since the population is greater than 10x the sample, the population is big enough.
4. Independence within Sample – Yes, the students’ responses are independent of each other.
Compute
Interpret
The pvalue we received was
Confidence Interval #1
The hypothesis test conducted above gives sufficient evidence that the majority of all Flagler College students find ongoing contact with both divorced parents important. Therefore, we will make a confidence interval to estimate the percentage of the population of all Flagler College students who believe it is important to have ongoing contact with divorced parents. Since we ran a onetailed test with a level of significance of 0.05, a 90% confidence interval will be created.
Prepare
1. Random Sample with Independent Observations – Though we are, again, not certain if the sample is truly random, we will make the assumption that it is in order to conduct our test. In addition, the students’ responses are collected independent of each other.
2. Large Sample – To check if the sample is large enough, we compute based on phat and check if the numbers for both expected successes and failures are larger than ten.
n*phat = (150)(0.82) = 123 > 10
n*(1 – phat) = (150)(1 – 0.82) = (150)(0.18) = 27 > 10
Since both expected successes and failures are more than ten, the sample is large enough.
3. Big Population – Flagler College’s population is around 2500 students. The population must be more than 10x bigger than the sample.
10n = (10)(150) = 1500 < 2500
Since the population is greater than 10x the sample, the population is big enough.
Compute
Interpret
We are 90% confident that between 76.84% and 87.16% of all Flagler College students believe that it is important for children of divorced parents to have ongoing contact with both parents. As both numbers in the interval are above 50%, this is the majority of all Flagler College students.
Hypothesis Test #2 Two Population Proportions
The contingency table below was created in order to compare the opinions of the students whose parents are Unmarried and Married regarding whether they think it is important or not to maintain contact with both divorced parents. Of the 67 students whose parents are not married, 49 of them said it is in fact important to maintain contact with both of them and of the 83 students whose parents are married, 74 said it is important to keep ongoing contact. That is, 73.1% (49 out of 67) of students whose parents aren't married, think it is important to maintain contact with them, versus 89.2% (74 out of 83) of students whose parents are still married think this contact is important. With an approximate difference of 16.1%, this sample gives us some reason to believe that the population of Flagler students whose parents are married and the population whose parents are not married, do differ in their opinions about this topic.
A hypothesis test will be used to determine if this difference is statistically significant for the population of students at Flagler College. This test will be done with a significance level of 0.05.
Hypothesize
Null: There is no difference in the proportion of the population of “Unmarried” students at Flagler and the proportion of the population of “Married” students at Flagler who feel ongoing contact with divorced parents is important.
Alternate: There is a difference in the proportion of the population of “Unmarried” students at Flagler and the proportion of the population of “Married” students at Flagler who feel ongoing contact with divorced parents is important.
Prepare

Large Samples It is found that the pooled sample proportion is
phat= (x1 + x2)/(n1 + n2) = ( 49 + 74 )/( 67+ 83)= 123/150= .82
Sample One (“Unmarried” students): since n1*phat= (67)(.82)= 54.94 > 10 and n1*(1phat)= (67) (1  0.82)= (67) (.18)= 12.06 > 10, sample one is large.
Sample Two (“Married” Students): since n2*phat= (83) (0.82)= 68.06 > 10 and n2*(1phat)= (83) (10.82)= (83) (.18)= 14.94> 10, sample two is large.
2. Random samples Probably not (but hoping they are representative). In order to proceed, we assume they are.
3. Independent Samples Yes, the students responses were taken in a way that the responses were independent of each other.
4. Independence between samples Yes, there is no relationship between the Unmarried students and the Married students.
Compute
Interpret
Since the pvalue is 0.0111, is less than our confidence level of 0.05, the null hypothesis will be rejected. Therefore, there is sufficient evidence that there exists a difference in the proportion of the population of “Unmarried” students at Flagler College and the “Married”students at Flagler College who feel ongoing contact with parents is important.
Confidence Interval #2 Estimate the Difference between Two Population Proportions
Thehypothesis test previously, gave us sufficient evidence that there is a significant difference in the opinion that ongoing contact with parents is important between the population of “Unmarried” students and “Married” students at Flagler College. Therefore a confidence interval will be created to estimate this difference and hopefully confirm that the two population proportions cannot be equal. Since a two tailed test with a level of significance of 0.05 was done, a 95% confidence interval will be performed.
Prepare

Random Samples with Independent Observations Probably not, but we hope it is representative. In order to proceed, we assume it is. In the other hand, the students responses were taken in a way that their responses were independent of each other.

Large Samples
Sample One (Unmarried students): since n1*phat1= (67)(.731)= 49 >10 and n1*(1phat1)= (67) (1 0.731)= (67) (.269)= 18 >10 , sample 1 is large.
Sample two (Married students): since n2*phat2= (83) (0.892)= 74>10 and n2*(1phat2)= (83) (10.892) = (83) (.108)= 8.96<10, sample 2 is not large enough.
3. Big Populations Flagler College has a population of approximately 2676 students. We are not sure what percentage of students think ongoing contact with parents is important. Because of this, we assume 50% think is important and 50% think is not. So, approximately (2676) (0.50)= 1338 students whose parents are married and (2676) (0.50)= 1338 students whose parents are divorced.
Population One (Unmarried students) Since (10)n1= (10) (67)= 670
Population Two (Married Students) Since (10)n2= (10) (83)= 830
4. Independent Samples Yes, the responses were taken in a way where their responses were independent of one another.
Compute
Interpret
The confidence interval is negative. This means that the percentage of the population of all Unmarried students who feel ongoing contact is important, is less than the percentage of the population of all Married students who feel this way. Therefore, we are 95% confident that the percentage of all Married students who feel ongoing contact is important is between 3.5% and 28.6% greater than the percentage of all Unmarried students who feel ongoing contact is important.
Conclusion
In today’s current social climate, divorce is an extremely common result within the United States. In this particular study, it was found that 47% of Flagler students have divorced parents, a quite substantial portion. This statistic seemed to influence the answers that were given to the question of whether students think that it is important for children of divorced parents to maintain contact with both parents. It was estimated that between 76.84% and 87.16% of students believe that maintaining contact with both parents was important, however, it was also estimated that children whose parents were still together were estimated to feel that ongoing contact is important between 3.5% and 28.6% more than the percentage of students whose parents were divorced. Perhaps this statistical difference can be attributed to children of divorced parents appreciating the nuances of the situation more due to the fact that they’ve experienced it themselves.
One sample proportion summary hypothesis test:p : Proportion of successes H_{0} : p = 0.5 H_{A} : p > 0.5 Hypothesis test results:

One sample proportion summary confidence interval:p : Proportion of successes Method: StandardWald 90% confidence interval results:

Contingency table results:
Rows: Parents Married Columns: Ongoing Contact Important
ChiSquare test:

Two sample proportion summary hypothesis test:p_{1} : proportion of successes for population 1 p_{2} : proportion of successes for population 2 p_{1}  p_{2} : Difference in proportions H_{0} : p_{1}  p_{2} = 0 H_{A} : p_{1}  p_{2} ≠ 0 Hypothesis test results:

Two sample proportion summary confidence interval:p_{1} : proportion of successes for population 1 p_{2} : proportion of successes for population 2 p_{1}  p_{2} : Difference in proportions 95% confidence interval results:

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