# Case Study Assignment

## Introduction

It has been necessary that the ratios of students to teachers be examined in order to find out whether more teachers should be employed to assist students or enroll more students to utilize the services provided by teacher. It has been found possible to find this relationship by the use of descriptive statistics to estimate whether more teachers should be employed or more students should be enrolled. The main method that has been identified is the use of statistical techniques such as the use of descriptive statistics: mean, median, standard deviation and variance as well as the use of analytical techniques such as correlation analysis. Correlation analysis has been done by determination of correlation coefficient.

Some of the schools in San Diego include Dana Middle School, Point Loma High School, San Diego Unified School and Herbert Hoover High School. This study involves the analysis of these schools in San Diego from the year 2000 until the year 2004.

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## Hypotheses for the Study

In order to understand the relationship between student population and teacher’s population, certain hypotheses were formulated. These hypotheses provided a tentative answer to the relationship between the number of teachers and student. Some of the hypotheses that were formulated include the following:

1. There is a positive correlation between the number of teachers and students in San Diego schools.
2. The ratio of students to teachers is used as a method of measuring class sizes.
3. Discussion.

It was assumed that a positive relation is where the ration of teachers to students does not exceed 20. A year where the student-teacher ratio is greater that 20 was assumed to be less supplied with the right number of teachers while a year with student-teachers ration below 20 was regarded to be under utilizing the teaching resources. The table below shows the student-teacher rations that were obtained from these years in San Diego Schools. The number of students is denoted by Y and the number of teachers is denoted by X. The values of X/Y represent the student teacher ratios in these years.

 Year X Y Y/X 2000 20 345 17.25 2001 25 422 16.88 2002 30 523 17.433 2003 23 345 15.0

From the students-teachers ratios above, it can be observed that all the ratios are less than 20. Therefore, between the years 2000 and 2004 there was no overpopulation of San Diego schools with students and teachers who were not overburdened with teaching.

SLO 1. Descriptive statistics

The data from students and teachers population were used to determine the mean, variance and standard deviations in order to obtain descriptive statistics for the analysis. The table below shows the procedures that were used.

 Year X Y X2 Y2 (X-Xbar) XY 2000 20 345 400 119025 -4.5 6900 2001 25 422 625 178084 0.5 10550 2002 30 523 900 273529 5.5 15690 2003 23 345 529 119025 -1.5 7935 Total 98 1635 2454 2299663 0 41075 Means 24.5 408.75 613.5 574915.75 median 24 383.5 – –

The following formula was used to estimate the standard deviation for the sample

∑(x-x bar)/ N)

=

According to the above data, the standard deviation equals to zero.

The data below was used to compute correlation coefficients for each of the relationships between the number of teachers and students

According to the above data, the following were the respective values for the components of the correlation coefficient. X2

4*41075 = 164300

Substituting these values into the above equation, we obtain the following values for r.

The value of correlation is 10,94%, which means the variable has a moderate positive correlation. This implies that there is a correlation between the number of teachers in the school and the number of students.

SLO 2. Calculation of probabilities of simple events using the understanding of dependent, independent and mutually exclusive events:

An example of dependent event: the probability of the teacher coming to school is 0,5 while the probability of him marking the register is also 0,5. The probability of marking the register is dependent on him coming to school. Hence the total probability = 0,5*0,5 =0,25.

An example of a mutually exclusive events: when the probability of rain occurring is 0,5; it means there will be no sunlight. Thus, the probability of sunlight is zero and the reverse is true.

An example of a complementary event: when the probability of a football team winning a match is 0,4, then the probability of the opposite team winning the match is 0,6. Since the sum of probabilities of complementary events is always equal to 1.

SLO 3. The use of normal distribution in hypothesis testing.

In the above analysis, the mean number of students was 408,75 while standard deviation was 0. When a sample of 10 students is selected from the students’ population, this population is called P. The population P is normally distributed if the mean of P = 408,75 and standard deviation of P is 0/A(10) = 0.048.

Hypothesis: The sample came from a population with mean of 408,75

In the case of 5% critical significance, the Z scores are between Z>-1,96 and Z< 1,96. The sample Z score is Z = (409-408,75)/0,048 = 5,208. Therefore, we accept the hypothesis at the 5% level of significance.

### Conclusion

The use of descriptive statistics can be useful in estimating the nature of relationship between variables. In the case of San Diego Schools, it has been possible to know the ration of students to teachers since the year 2000. Furthermore, this information can be used in planning the recruitment of new teachers and enrollment of more students into the school. It is also a method of knowing the correlation between variables. For instance, it has been possible to know the correlation between the number of students and teachers.