Math 153 Statistics For The Assessment Answers

From the above scatterplot, it is apparent that the relationship between the two variables is positive and also it is strong as the deviation of the points from the line of best fit is moderate only. 

Regression equation  

%LIWF = 15.63 + (1.39* percent of 18-64 years old with no HS diploma) 

R square value (From scatter plot) = 0.4960

The value indicates that 49.60% of variation in percent of low-income working families would be explained by variation in percent of 18-64 years old with no HS diploma.  

Correlation coefficient (R) =0.7043 

The correlation coefficient comes out to be positive which indicates that when there is decrease in the percent of 18-64 years old with no high school diploma will no doubt lead to decrease in percent of low-income working families. 

In order to collect a simple random sample of 20 jurisdictions, it makes sense that on a random basis 20 jurisdictions are selected from the given 51 jurisdictions. The data corresponding to these jurisdictions may be taken as the requisite sample data.  

Sample mean and sample standard deviation of percent of low-income working families is computed and is shown below. 

It is unlikely that two sample would produce the same sample mean or standard deviation since all the jurisdictions are different and have dissimilar characteristics with regards to the given variables of interest. 

90% confidence interval

The t value for 90% confidence interval = 1.73

Lower value of 90% confidence interval = Mean – (t value *Standard error)

Upper value of 90% confidence interval = Mean + (t value *Standard error)

Mean = 31.65

Standard error = standard deviation/sqrt (sample size)

Standard deviation = 7.57

Sample size = 20

Standard error = 7.57/sqrt (20) = 1.69

Now,

Lower value of 90% confidence interval = Mean – (t value* Standard error) =31.65 – 1.69 =28.7231

Upper value of 90% confidence interval = Mean + (t value *Standard error) = 31.65 +1.69 = 34.5769

90% confidence interval = [28.7231   34.5769] 

It can be concluded with 90% confidence that average percent of low income working families for all the jurisdictions (i.e. population) lies between 28.72% and 34.57%. 

The interval with lower confidence interval would not be advantageous to the jurisdictions as a lower confidence interval would be even narrower and therefore lesser jurisdictions would be eligible for federal funds. It would be in the best interest of the jurisdictions that a higher confidence interval is considered which would ensure that more jurisdictions are included. 

The reporting of research results for obtaining of Federal aid for the concerned jurisdiction would amount of misuse of statistics as the results would be biased and hence would not be valid. It makes sense that the research results should be obtained by a team which comprises of the representation from different jurisdictions selected at random so as to minimise jurisdiction based bias. 

Hypothesized population mean =31.3%

Sample mean =29.8%

Sample size = 16

Sample standard deviation = 4.1%

One sample t test is considered appropriate test for testing the claim.

Assumption

Sample data follows normal distribution

Population is normally distributed

Variable which has been measured would be wither ration or interval

Sample should be randomly selected 

Hypothesis testing

Null hypothesis H₀:  i.e. the average percent of low-income working families in 2014 does not differ from the corresponding value in 2011.

Alternative hypothesis H_a  i.e. the average percent of low-income working families in 2014 is lower than the corresponding value in 2011. 

Test statistic (z score, t score)

Sample size = 16

Sample proportion = 29.8%

Sample standard deviation = 4.1% 

The t score = -1.463

Degree of freedom = 16-1 = 15 

The p value corresponding to t score and degree of freedom = 0.09

Level of significance = 0.10

It can be seen from the above that p value is lower than level of significance and hence, sufficient evidence is present to reject the null hypothesis and to accept the alternative hypothesis. 

The conclusion can be made that national average percent of low-income working families has improved since 2011.