Math 10 Extra Credit Sample - Education and Income
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University of California, Berkeley *
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MISC
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Mathematics
Date
Apr 3, 2024
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7
Uploaded by ChiefHawk1769
1 Is earning a College degree really worth it? FirstName LastName Math 10 Fall 2015 De Anza College
2 Brian Stoffel, author of the article, “
The Average American Made This Much Last Year, by Level of Education
—
How Do You Compare?
”
depicts the correlation between education and average annual earnings. According to Stoffel, “Throwing your hat in the air can mean a significant bump in income.” Stoffel claims that those w
ho earn a higher level of education average a higher level of annual household income. He poses the following question to his readers, “Is it worth it?” Stoffel provides the reader with data from a Consumer Expenditure Survey conducted by the Bureau of Labor Statistics further supporting his claim. Level of Education Average Income Less than high school $28,000 High school graduate $40,000 Some college $48,000 Associate degree $61,000 Bachelor degree $85,000 Masters, PhD (Professional) $124,000 The results indicate those who have less than a high school diploma earn on average $28,000 per year. Those who graduate from high school earn on average $40,000 per year. Students who select to complete a Bachelor Degree earn $24,000 per year more than those with an Associate degree. Lastly, individuals who earn the highest amount of income are those who complete a Masters or PhD degree at $124,000 per year. I would agree with Stoffel that there is a correlation between the level of education a person achieves and the amount of average income he/she is expected to earn annually. According to the May 2014 State Occupational Employment and Wage Estimates California by the Bureau of Labor Statistics, the annual income for a lawyer is $158,200. Lawyers must have a JD degree and pass the Bar exam. In contrast, a social and human service assistant requires only a high school diploma hence the annual average income is only $35,310. My personal observations and experience also lead me to believe that many careers today such as engineers, physicians, pharmacists and dentists require a higher level of skill compared to those who may
3 choose a career working in a fast food restaurant. The higher the level of skill required, the more education is needed. In the article, Stoffel does not address the outliers of the correlation. These are the billionaires of today that dropped out of college and became entrepreneurs. Ralph Lauren, Mark Zuckerberg, Larry Ellison, and Michael Dell to name a few. Stoffel also does not include that there is a lower unemployment rate amongst those with a higher level of education than those with a lower level of education. According to the Bureau of Labor Statistics in 2014, those with a high school diploma averaged an unemployment rate of 9% while those with a Bachelor degree averaged a rate of only 3.5%. Although Stoffel presents the reader with sufficient evidence to support his claim, I have decided to conduct my own experiment. The experiment conducted will be fictional. The two factors in the experiment are level of education and income. The level of education is the independent variable and the income is the dependent variable. I am interested in knowing if the annual income a person earns is dependent on the level of his/her education. I am hoping to learn if it is more likely that a person with a higher level of education will earn a higher level of income. When conducting the experiment, I will use stratified random sampling. In a stratified random sample, a population is divided into subgroups, called strata and a sample is randomly selected from each stratum. In this experiment, a total of 1,364 students from Fremont High School in Cupertino, CA are sampled about their level of education achieved and their annual income three years, five years and seven years following graduation from Fremont High School. The population is divided into four groups based on their level of education: High School Dipl
oma, Associate Degree, Bachelor Degree, and Master’s Degree/PhD. Annual income is divided into four subgroups: $40K - $48K; $48K - $61K; $61K - $85K; $85K-$124K Letters will be mailed out to the participants requesting the information. A problem that may occur with this method causing a bias would be participants who choose not to respond or refuse to participate. One way to help alleviate this bias is to ensure the participant that the results are confidential and that the results will only be used strictly for the study.
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4 Additional information that may be relevant to the study is the loss of contact with the participants due to unforeseen circumstances such as the participant moving out of the country. The observed values or data values from the experiment are depicted in the following contingency table. The expected values or the values I would expect to obtain for the experiment if the null hypothesis were true are as follows: Level of Education $40,000-
$48,000 $48,000-
$61,000 $61,000-
$85,000 $85,000-
$124,000 Row Total High School Diploma 110 96 45 15 266 Associate Degree 100 132 92 36 360 Bachelor Degree 86 97 150 102 435 Master’s Degree/PhD 34 40 90 139 303 Column Total 330 365 377 292 1364 Level of Education $40,000-
$48,000 $48,000-
$61,000 $61,000-
$85,000 $85,000-
$124,000 High School Diploma (E1) 64.355 (E2) 71.18 (E3) 73.521 (E4) 56.944 Associate Degree (E5) 87.097 (E6) 96.334 (E7) 99.501 (E8) 77.067 Bachelor Degree (E9) 105.24 (E10) 116.4 (E11) 120.23 (E12) 93.123 Master’s Degree/PhD (E13) 73.386 (E14) 81.081 (E15) 83.747 (E16) 64.865
5 The equation for the test statistic for an independence test can be written as: (O-E)
2 X
2 = E (i-j) O is the observed values and E is the expected values. The expected value for each cell needs to be at least five in order for the independence test to be used. The degrees of freedom = (j-1)(i-1) where j is the number of columns and I is the number of rows. A test for independence determines whether two factors are independent or not. Expected values were obtained using the following formula: E1= (Total Row 1) (Total Column 1) Total Number Surveyed E1 = (266)(330) = 87,780 = 64.355 1364 1,364 E14 = (Total Row 4)(Total Column 2) Total Number Surveyed E14 = (303)(365) = 110,595 = 81.081 1364 1,364 E7 = (Total Row 2)(Total Column 3) Total Number Surveyed E7 = (360)(377) = 135,720 = 99.501 1364 1364 When conducting the study, the null hypothesis is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt. The null hypothesis is denoted as H
o
or H naught. The alternative hypothesis is a claim about the population that contradicts
6 H
o and what we will conclude if we decide to reject H
o
. The alternative hypothesis is denoted as H
a
or H sub a.
H
o
: The level of education is independent of the level of income.
H
a
: The level of education is dependent upon the level of income.
5% significance level will be used.
Chi-squared distribution will be used X
2 = (O-E)
2 The Chi-squared distribution is used to find relationships between independence. A test for independence determines whether two factors are independent or not. For example, does the level of education affect the level of income he/she earns? In this study, X
2
= 262.6104196 = 262.6104 The degrees of freedom determine how the chi-square curve looks. It is nonsymmetrical and skewed to the right. The u
= df df = (number of columns-1)(number of rows-1) df = (4-1)(4-1) df = (3)(3) df = 9 X = X
2
9
The p-value is the probability that if the null hypothesis is true, the results from another randomly selected sample will be as extreme (or more extreme) as the results obtained from the given sample. A large p-value indicates that we should not reject H
o
. The smaller p-value, the more unlikely the outcome, and the stronger the evidence against the null hypothesis. We would reject the null hypothesis if the evidence is strongly against it. P-value = X
2
cdf(262.6104196,10^
10
, 9) = 2.162003 x 10
-51
In this case, the p-value is extremely small, 2.162003 x 10
-51
, so the outcome is very unlikely and the stronger the evidence is against the null hypothesis. 16 1 E
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7 Conclusion: Since the p-value < alpha, where alpha = .05, we reject H
o . At the 5% significance level, there is sufficient evidence to conclude that the level of income is dependent on the level of education. The study supports the findings of the article and proves that the level of income is dependent upon the level of education. The higher the level of education a person has, the higher their level of income. Although the information in this study and article were very informative, one aspect of the study I would change is to add a third variable. I feel that the level of unemployment associated with the level of education a person has may also be helpful to the reader. I believe one flaw of this study is that the results give the reader the impression that if you obtain a higher level of education, you will automatically earn a higher level of income. Stoffel somewhat clarifies this by stating in the article, “The income of each group represents the mean, or average. That means that those who are high earners---think CEOs
—
disproportionately bring the average up. Those who are older do the same thing, as they are further along t
heir career earnings arc.”
In regards to the overall project, I feel that I have gained an understanding of how statistical data can help you make decisions. It has given me the insight that the benefit of a higher education far outweighs the costs. I would agree with Stoffel in that “
the benefits of a degree are astounding over a lifetime
.”
p-value 2.162003 x 10
-51
9 262.1604