Introduction
In this paper we will describe and discuss the application of probability and statistics to a real world problem. The problem we discussing is about substance abuse problem at universities and colleges. The problem of prevalence of substance abuse among young people attracts the attention of researchers worldwide. Some of the effective methods of studying the prevalence of drug use or alcohol consumption is an anonymous survey organized by various groups, including students' general educational institutions. At the same time, the problem of consistently high incidence of alcoholism and medical and social consequences of alcohol abuse among young adults is also being discussed. Therefore, the study data on the prevalence and incidence of substance abuse among young adults does not lose its relevance. According to the conditions of the assignment, we are informed that the national average percentage of involving in some type of substance abuse in the United States of America is 46%.
The goal of this paper is to examine the issue of substance abuse at Rocky University. Some members in the university have serious concerns and believe that substance abuse at Rocky University is even more widespread problem than in other universities. Others think that the level of substance abuse among the youth of Rocky University is not higher than national average, thus, it is not the first problem of the college. Our task is to evaluate the significance of the problem by using descriptive statistics and statistical inference using the given data.
We will separately consider the levels of participating in binge drinking, abusing prescription medication and the level of illegal or illicit drugs use. Confidence intervals for the proportion of all students who were involved in these types of substance abuse will be developed. After this, the level of substance abuse will be compared to the national average in order to find out whether the problem of substance abuse at Rocky University is a major problem or not.
Data and Variables
For the purposes of our research we are given with the data set of 90 observations. These observations represent the results of anonymous survey conducted among the sample of 90 students from this year’s graduating class. They have answered the following questions:
During your time at Rocky, did you ever participate in binge drinking? (Yes / No)
During your time at Rocky, did you ever abuse prescription medication? (Yes / No)
During your time at Rocky, did you ever use any type of illegal or illicit drug (not included in question #1 and question #2)? (Yes / No)
The answers of the students were recorded by researchers. If a student has answered “Yes” on at least one question, he was considered as an individual involved in substance abuse of some type.
For the convenience of the statistical computations, we recode the answers of the students in the following way: Yes = 1, No = 0. The data set consists of the following variables:
Involved in Binge Drinking – whether a student involved in binge drinking or not.
Abused Prescription Med. or Marijuana – whether a student abused prescription medication or not
Used Illegal Drugs – whether a student used illegal or illicit drugs (not included in previous two questions) or not.
Gender – the gender of a student (Male or Female)
Substance Abuse (in general) – whether a student answered “Yes” on at least one question or not.
Descriptive Statistics
The first step of this research is to give more information about the given data. We develop descriptive statistics in order to understand the main characteristics of distributions of the variables. Since all the variables are dichotomous, we conclude that the distribution of each variable is binomial (Stattrek.com, 2016). We separate the data by gender and provide the following descriptive statistics:
For females (number of observations = 44):
For males (number of observations = 46):
There were 44 females and 46 males participating in the study. Nine females were involved in binge drinking (20.45%), thirteen were abused prescription medication or marijuana (29.55%) and fifteen used illegal drugs (34.09%). Generally, 24 females were involved in some type of substance abuse during their time at Rocky University (54.55%).
Thirteen males were involved in binge drinking (28.26%), twelve were abused prescription medication or marijuana (26.09%) and ten used illegal drugs (21.74%). Generally, 23 males were involved in some type of substance abuse during their time at Rocky University (50%).
The distribution of positive and negative answers and the distribution of the proportion of substance abuse among the 90 students is visualized on the following histograms of frequency and proportion distribution:
We can see that in percentage notation, females are less likely to be involved in binge drinking than males. However, females are more likely to abuse prescription medication or marijuana, use illegal or illicit drugs. The overall level of substance abuse of females is higher than of males.
Inferential Statistics and Findings
In this chapter we will develop confidence intervals for proportions on the different types of substance abuse and also conduct a hypothesis test to determine of the proportion of all students at Rocky University who were not involved in some type of substance abuse is less than that of all U.S. students elsewhere.
We develop 90% confidence interval for all the three types of substance abuse and for the overall proportion of substance abuse for males and females separately. We know that 90% confidence interval can be calculated by using the following formula (Stattrek.com, 2016):
p±1.645p1-pN
where p is a sample proportion, N is the number of observations in the sample, 1.645 is a 90% confidence z-score.
The results of the calculations of confidence intervals are given in the tables below:
We are 90% confident that:
The true population proportion of females who were involved in binge drinking is between 10.45% and 30.46%
The true population proportion of females who abused prescription medication or marijuana is between 18.23% and 40.86%
The true population proportion of females who used illegal or illicit drugs is between 22.34% and 45.85%
The true population proportion of the overall level of females’ involving in any kind of substance abuse is between 42.20% and 66.89%
The true population proportion of males who were involved in binge drinking is between 17.33% and 39.18%
The true population proportion of males who abused prescription medication or marijuana is between 15.44% and 36.74%
The true population proportion of males who used illegal or illicit drugs is between 11.73% and 31.74%
The true population proportion of the overall level of males’ involving in any kind of substance abuse is between 37.87% and 62.13%
The next step is to conduct a hypothesis test to examine the proportion of all students at Rocky University who were not involved in some type of substance abuse. We want to check the claim that this level is less than that of all the U.S. students elsewhere. The national level of involving in substance abuse is 46%, hence, the percentage of those who were not involved is 54%.
Set the pair of statistical hypotheses.
Null hypothesis: the population proportion of students at Rocky University who were not involved in any type of substance abuse is not significantly different from 0.54.
Alternative hypothesis: the population proportion of students at Rocky University who were not involved in any type of substance abuse is significantly less than 0.54.
H0:p=0.54Ha:p<0.54
Set level of significance alpha at 0.10.
Use one-sample z-test for proportions (Statisticslectures.com, 2016):
z=p-p0p01-p0n
The sample proportion of all students who weren’t involved in some kind of substance abuse is:
p=20+2390=4390=0.4778
z=0.4778-0.540.541-0.5490=-1.184
z-critical for 90% confidence level is 1.645.
Since 1.184<1.645, we failed to reject the null hypothesis.
Conclusion
In this assignment we investigated substance abuse problem at Rocky University. It is appeared that females are less likely to be involved in binge drinking than males. However, females are more likely to abuse prescription medication or marijuana, use illegal or illicit drugs. The overall level of substance abuse of females is higher than of males.
The proportion of students at Rocky University who were not involved in any type of substance abuse is not significantly different from national average. That is why we would like to say that this is not a major issue in the college. The study can be improved if researchers will examine other possible factors that may have a significant impact on involving in various kinds of substance abuse. These factors may be student’s performance, family status, family income, health (physical or mental) status and so on.
References
Statisticslectures.com. (2016). One Sample z-Test for Proportions. Retrieved 24 January 2016, from http://www.statisticslectures.com/topics/onesamplezproportions/
Stattrek.com. (2016). Binomial Calculator. Retrieved 24 January 2016, from http://stattrek.com/online-calculator/binomial.aspx
Stattrek.com. (2016). Proportion: Confidence Interval. Retrieved 24 January 2016, from http://stattrek.com/estimation/confidence-interval-proportion.aspx?Tutorial=AP
