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Phase Three: Students and Plus/Minus Grades

Introduction:

On the first phase of this project, a targeted population of 150 Flagler students from the MAT 223 Statistics classes were surveyed about Plus/Minus Grades. In the second phase of this report, this same sample of 150 students was divided into two smaller groups. The two sample sizes are the sample of Flagler College students who are male and female who were surveyed on their knowledge of Plus/Minus Grades. There are 110 students sampled with no knowledge of plus-minus grades, and 40 students sampled with knowledge of plus minus grades. A bar chart representing the two samples is presented below.

Result 1: STUDENTS SAMPLED FREQ TABLE   [Info]

In this phase of the report, attention will be given to students' opinions on whether or not a Plus/Minus grading system should be kept in use at Flagler College.

First, methods of statistical inference will be used to determine if the sample results indicate that the majority of the population of all Flagler College students feel that a Plus/Minus grading system should be kept in use at Flagler College. A hypothesis test will first be run to find statistical evidence of majority and then a confidence interval will be created to estimate the percentage of the population at Flagler College students who feel a Plus/Minus grading system should be kept at Flagler College.

Second, the sample results will also be used to determine if the opinion of the population of all students who know of the Plus/Minus system and the population of all students who do not know of the Plus/Minus grading system at Flagler College have a staticatically significant difference of opinion regarding whether a Plus/Minus grading system should be kept at Flagler College. Again, a hypothesis test will be run to find statistical evidence of a difference and then a confidence interval will be created to estimate the difference in the percentage of the population of students with knowledge of the Plus/Minus grading system and students without knowledge of the Plus/Minus grading system who feel a Plus/Minus system should be kept at Flagler College.

Hypothesis Test 1

In the sample of 150 students, 107 reported that a Plus/Minus grading system should be kept at Flagler College.  That is, the majority, 71.33%, of the students sampled expressed that a Plus/Minus grading system should be kept at Flagler College. These sample results will be used to test the claim that the majority of the population of Flagler College students who believe that a Plus/Minus grading system should be kept at Flagler College at a level of significance of 0.05  A pie chart of the data is given below.

Result 2: Pie Chart With Data PHASE 3   [Info]

Hypothesize

Null: 50% of all Flagler College students believe that a Plus/Minus grading system should be kept.

Alternate: More than 50% of all Flagler College students believe people should be allowed to grow small quantities in their homes.

Based on the alternate hypothesis, this is a right-sided test.

Prepare

1. Random Sample – Frankly, probably not (but we hope it is representative).  However, to proceed, we will assume it is.

2. Large Sample – Since np0 = (150) (0.50) = 75 > 10 and n(1-p0) = (150) (0.50) = 75 > 10 are both true statements, the sample is large.

3. Big Population – Since 10n = (10)(150) = 1500 < 2500, the population is big.  Recall, Flagler College has a population of appropriately 2500 students.

4. Independence within Sample – Yes, the student responses were taken in such a way that their responses were independent of each other.

Compute

Result 3: One sample proportion summary hypothesis test PHSE 33   [Info]

### One sample proportion summary hypothesis test:

p : Proportion of successes
H0 : p = 0.5
HA : p > 0.5

Hypothesis test results:
ProportionCountTotalSample Prop.Std. Err.Z-StatP-value
p1071500.713333330.0408248295.2255781<0.0001

Interpret

Since the p-value (0.0001) is less than the level of significance of 0.05, the null hypothesis must be rejected.  Therefore, there is sufficient evidence to support the claim that the majority of all Flagler College students feel that a Plus/Miunus grading system should be kept at Flagler College.

Confidence Interval #1 – Estimating the Population Proportion

The hypothesis test gives sufficient evidence that the majority of all Flagler College students feel that a Plus/Minus grading system should be kept. Therefore, a confidence interval will be created to estimate the percent of the population of all Flagler College students who believe that a Plus/Minus grading system should be kept. Since a one tailed test with a level of significance of 0.05 was run, a 90% confidence interval will be created.

Prepare

1. Random Sample with Independent Observations – Again, probably not (but we hope it is representative).  However, to proceed, we will assume it is.  Furthermore, yes, the student responses were taken in such a way that their responses were independent of each other.

2. Large Sample – Since n*phat = (150)(0.7133) = 107 > 10 and n*(1 – phat) = (150)(1 – 0.7133) = (150)(0.2867) = 43 > 10, the sample is large.

3. Big Population – Since 10n = (10)(150) = 1500 < 2500, the population is big.  Recall, Flagler College has a population of appropriately 2500 students.

Compute

Result 4: One sample proportion summary confidence interval PHASE 3   [Info]

### One sample proportion summary confidence interval:

p : Proportion of successes
Method: Standard-Wald

90% confidence interval results:
ProportionCountTotalSample Prop.Std. Err.L. LimitU. Limit
p1071500.713333330.0369223410.652601490.77406518

Interpret

We are 90% confident that between 65.3% and 77.4% of all Flagler College students feel that a Plus/Minus grading system should be kept. This is certainly the majority of all Flagler College students.

Hypothesis Test #2 – A Claim of the Difference between two Population Proportions

Result 5: Contingency table (with data) PHASE 3   [Info]

### Contingency table results:

Columns: Keep Plus-Minus
NoYesTotal
No3872110
Yes53540
Total43107150

### Chi-Square test:

StatisticDFValueP-value
Chi-square16.97155760.0083

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 run at a level of significance of 0.05.

Hypothesize

Null: There is no difference in the proportion of the populations of students at Flagler College who did and did not know about the Plus/Minus system who believe that Flagler College should keep a Plus/Minus grading system.

Alternate: There is a difference in the proportion of the populations of students at Flagler College who did and did not know about the Plus/Minus system who believe that Flagler College should keep a Plus/Minus grading system.

Based on the alternate hypothesis, this is a two tailed test.

Prepare:

1.      Large Samples – It is found that the pooled sample proportion is

p-hat = (x1 + x2)/(n1 + n2) = (72 + 35)/(110 + 40) = 107/150 = .7133

Sample One (students who knew about the Plus/Minus system): Since n1*p-hat = (110)(.7133) = 78.463 > 10 and

n1*(1 - p-hat) = (110)(1 – 0.7133) = (110)(0.2867) = 31.537 > 10, sample one is large.

Sample Two (students who did not know about the Plus/Minus system): Since n2*p-hat = (40)(0.7133) = 28.532 > 10 and

n2*(1 - p-hat) = (40)(1 – 0.7133) = (40)(0.2867) = 11.468 > 10, sample two is just large enough.

2. Random Samples – Again, probably not (but we hope they are representative).  However, to proceed, we will assume they are.

3. Independent Samples – Yes, the student responses were taken in such a way that their responses were independent of each other.

4. Independence between Samples – Yes, there is no relationship between the students who knew about the Plus/Minus system and those who did not.

Result 6: Two sample proportion summary hypothesis test PHASE 3   [Info]

### Two sample proportion summary hypothesis test:

p1 : proportion of successes for population 1
p2 : proportion of successes for population 2
p1 - p2 : Difference in proportions
H0 : p1 - p2 = 0
HA : p1 - p2 ≠ 0

Hypothesis test results:
DifferenceCount1Total1Count2Total2Sample Diff.Std. Err.Z-StatP-value
p1 - p2721103540-0.220454550.083493785-2.64037070.0083

Since the p - value = 0.0083 is less than the level of significance 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 students who knew about the Plus/Minus system and the proportion of the population of students who did not know about the Plus/Minus system at Flagler College who believe that a Plus/Minus system should be kept.

Confidence Interval #2 –Estimate the Difference between two Population Proportions

The hypothesis test gave us sufficient evidence that there is a significant difference in the opinion that Flagler College should keep a Plus/Minus grading system between the populations students at Flagler College who did and did not know about the system. 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 run, a 95% confidence interval will be created.

Prepare

1. Random Samples with Independent Observations – Again, probably not (but we hope it is representative).  However, to proceed, we will assume it is. Furthermore, yes, the student responses were taken in such a way that their responses were independent of each other.

2. Large Samples –

Sample One (students who knew about the Plus/Minus system): Since n1*p-hat1 = (82)(0.58) = 47.56 > 10 and

n1*(1 - p-hat1) = (110)(1 – 0.6545) = (110)(0.3455) = 38.05 > 10, sample one is large.

Sample Two (students who did not know about the Plus/Minus system): Since n2*p-hat2 = (40)(0.875) = 35 > 10 and

n2*(1 - p-hat2) = (110)(1 – 0.875) = (110)(0.125) = 13.75 > 10, sample two is large.

3. Big Populations – Recall, Flagler College has a population of appropriately 2500 students.  Since we are unsure what overall percentage of the students are or are not affected by the grading system, we will assume 50% are and 50% are not.  Hence, there are approximately (0.50)(2500) = 1250 students who are affected and (0.50)(2500) = 1250 students who are unaffected in the population.

Population One (students who knew about the Plus/Minus system): Since 10n1 = (10)(110) = 1100 < 1250, population one is big.

Population Two (students who have not smoked marijuana): Since 10n2 = (10)(40) = 400 < 1250, population two is big.

4. Independent Samples – Yes, the student responses were taken in such a way that their responses were independent of each other.

Compute

Result 7: Two sample proportion summary confidence interval PHASE 3 R7   [Info]

### Two sample proportion summary confidence interval:

p1 : proportion of successes for population 1
p2 : proportion of successes for population 2
p1 - p2 : Difference in proportions

95% confidence interval results:
DifferenceCount1Total1Count2Total2Sample Diff.Std. Err.L. LimitU. Limit
p1 - p2721103540-0.220454550.069209626-0.35610292-0.084806172

Interpret

This confidence interval is completely negative; this indicates that the percentage of the population of all students who knew about the Plus/Minus system who feel that Flagler College should keep the system is less thab the percentage of the population of all students who did not know about the Plus/Minus system who feel that Flagler College should keep the system. Thus, I am 95% confident that the percentage of all students who did not know about the Plus/Minus system who feel that Flagler College should keep a Plus/Minus grading system is between 8.48% and 35.6% greater than the percentage of student who did know about the Plus/Minus system who feel that Flagler College should keep a Plus/Minus grading system.

Conclusion

Plus/Minus grading systems have become a popular topic among students and the acceptance of Flagler students towards one such system is an important topic. In this report, the sample provided evidence that the majority of all Flagler College students believe that a Plus/Minus grading system should be kept.  In fact, it was estimated that between 65.3% and 77.4% of all Flagler College students feel that a Plus/Minus grading system should be kept. Furthermore, it was found that there is statistical evidence that those students who knew about the grading system have a higer percentage of believing that Flagler College should keep a Plus/Minus grading system. It was estimated that between 8.48% and 35.6% less than all Flagler College students who did not know about the system than those Flagler College students who did.

The underlying purpose of a Plus/Minus grading system is to encourage students to work harder for their grades, while also providing a more accurate grading scale. Therefore, it is no surprise that most students feel that Flagler College should keep one such system. Furthermore, this grading system is what most students have been graded with their whole academic careers, so it is normal for many. A Plus/Minus grading system makes sense and is practical, therefore it is no surprise that most students want to keep it.