Part 1: Correlation and Linear Regression
There is a population of workers in this country who are employed but who do not earn enough in wages to securely cover the cost of their family’s basic needs. Although the numbers vary by state or district, these individuals and their families are similar in that they cross racial, cultural, and local boundaries. They are considered the nation’s “Working Poor.” This common graded assignment was designed to allow you, the Introductory Statistics student, to explore some data that may be related to this population. Through the use of multiple statistical concepts covered in your MATH 153 course you will complete the provided assignment below to demonstrate your understanding of the manipulation of data, the use of technology in statistical analysis, the relationships between related variables, estimating population parameters, and hypothesis testing.
To assess your understanding of selected key statistical concepts through the use of real-world data related to the working poor in the U.S.
Present responses as if addressing individuals with a basic but thorough mastery of the appropriate statistical concepts and procedures required for the completion of this assignment.
Please complete this assignment by responding specifically to each of the items 1-15, provided. According to the instructions you will be expected to use technology to make calculations, to create and include graphical displays, and to support your responses with complete, well-organized, and clear statements and interpretations.
It is widely believed that the more education one receives the higher the income earned at the time of first employment and over the course of a career. However, due to varying reasons, many people never complete high school and, thus, never receive their high-school diploma. Although individuals without a high-school diploma are often able to find employment, they experience economic outcomes quite different from those who finish high school before entering the workforce to earn a living. Across the nation, there are millions of individuals with families who are now working but do not possess the credentials of a high-school diploma. Many of these individuals and their families are considered to be a part of the working poor that make up a considerable portion of this nation’s labor force.
1.Use technology to create and provide a scatterplot of the association between the “percent of low-income working families” and the “percent of 18-64 yr-olds with no high school diploma” data for each jurisdiction. Write at least two sentences explaining how/why it is appropriate to create such a scatterplot, and describe the characteristics of the association seen in the scatterplot. Be sure to use the actual names of the variables in their appropriate places in your response(s). (Print or copy-and-paste the scatterplot and be sure to clearly identify the predictor and response variables based on the possible believed association.)
Part 2: Confidence Intervals
2.Use technology to find the regression equation for the linear association between the “percent of low-income working families” and the “percent of 18-64 yr-olds with no high school diploma.” (Round final values to two decimal places.) Provide this equation and write a brief interpretation of the slope using the variable names. (Print or copy-and-paste the printout that identified the equation of the linear regression line, or any other form of evidence that technology was used.)
3.A student states that a decrease in the “percent of 18-64 yr-olds with no high school diploma” will lead to a decrease in the “percent of low-income working families.” Write at least two concise sentences addressing the key uses of linear correlation and comment on its limitations in a response to the student’s statement
4.Calculate and provide the R-squared value for the regression equation. Provide a statement about its meaning, in general, and, its specific interpretation in the context of this assignment
5.After examining these data for all the jurisdictions, someone notes that certain areas have an unusually high “percent of 18-64 yr-olds with no high school diploma.” Based on this finding, this individual concludes that the high percentages are due to the rising population of immigrants in those areas. Further, the individual argues that any estimates of the associated “percent of low-income working families” in those areas should be recalculated after removing this sub-population from the data set, as they are causing the area to “look bad”. In addition to thinking critically, use the key rules about linear regression and extrapolation to write a statistically appropriate and socially responsible response to the individual’s conclusion and argument.
During the recovery from the Great Recession of 2007-2009, the economic situation for many families improved. However, in 2011 the recovery was slow and it was uncertain as to how much had really changed on the national level. To estimate the national average of the percent of low-income working families, a representative simple random sample of the percent of low-income working families from each of the country’s reporting jurisdictions could be used to calculate a point estimate and create a related confidence interval. With this confidence interval a better picture of the nation’s recovery can be had and legislative decisions can be made.
6.Describe in two or three sentences how a simple random sample of size n=20 could be obtained from the full list of jurisdictions provided for use with this assignment.
Part 3: Hypothesis Testing
7.A researcher reported that a sample of size n=30 produced a sample mean of 32.56% and a sample standard deviation of 6.56%. Use this information to calculate a 90% confidence interval for the national average for the “percent of low-income working families”. Provide the upper and lower limits of the confidence interval and the margin of error. (Round the limits to two decimal places.)
8.Provide an explanation as to why it would be very unlikely that a different sample of size n=30 would produce the same confidence interval.
9.Provide an appropriate statistical interpretation of the 90% confidence interval found in number 7.
10.If a limited amount of federal funds have been allocated to assist jurisdictions whose “percent of low-income working families” exceeds a threshold based on the upper limit of a confidence interval, what would be the effect of using a confidence level that is higher than 90%?
11.If a public official requests funds based on a confidence interval provided by constituents in his/her district, would this raise any ethical concerns, or constitute a misuse of statistics, or both? Provide at least two sentences to respond to the situation presented
In 2011, the national percent of low-income working families had an approximately normal distribution with a mean of 31.3% and a standard deviation of 6.2% (The Working Poor Families Project, 2011). Although it remained slow, some politicians claimed that the recovery from the Great Recession was steady and noticeable. As a result, it was believed that the national percent of low-income working families was significantly lower in 2014 than it was in 2011. To support this belief, a spring 2014 sample of n=16 jurisdictions produced a sample mean of 29.8% for the percent of low-income working families, with a sample standard deviation of 4.1%. Using α=0.10 significance level, test the claim that the national average percent of low-income working families had improved by 2014.
12.Clearly restate the claim associated with this test, and state the null and alternate hypotheses.
13.Provide two or three sentences to state the type of test that should be performed based on the hypotheses. Additionally, state the assumptions and conditions that justify the appropriateness of the test.
14.Use technology to identify, and then provide the test statistic and the resulting P-value associated with the given sample results. Provide a statement that explains the interpretation of the P-value. (Print or copy-and-paste the output that identified these values, or any other form of evidence that technology was used.)
15.State, separately, both the decision/result of the hypothesis test, and the appropriate conclusion/statement about the claim.