Question

A neighborhood council is interested in the family income and medical care expenditures of its community. In particular, it is believed that income is related to to medical care expenditures. Below are family income (per 1,000 dollars) and medical care expenditure (per 100 dollars) data from a random sample of households in the community. What can be concluded with α = 0.05?

family income	medical care
8 5 9 11 13 16 17 18 18 21	21 16 18 13 12 15 7 8 2 8

a) What is the appropriate statistic?
---Select--- na Correlation Slope Chi-Square
Compute the statistic selected above:  

b) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

c) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
effect size =  ;   ---Select--- na trivial effect small effect medium effect large effect

d) Make an interpretation based on the results.

There was a significant positive relationship between family income and medical care expenditures.There was a significant negative relationship between family income and medical care expenditures.    There was no significant relationship between family income and medical care expenditures.

Answer 1

Answer a) The appropriate statistic is correlation

Explanation:

We have selected correlation because it is a technique for investigating the relationship between two quantitative, continuous variables

Answer b)

In this case correlation coefficient (r) is the test statistic. Following steps are following to calculate correlation coefficient:

Thus, the test statistic r = -0.7977

The sample size is n = 10, so then the number of degrees of freedom is df = n - 2 = 10 - 2 = 8

The corresponding critical correlation value r_c​ for a significance level of α = 0.05, for a two-tailed test is:

r_c​ = 0.632 (Obtained using table of critical value for the correlation coefficient)

The following hypothesis needs to be tested:

H_0​: ρ = 0 (Correlation coefficient is not significant)

H_a​:ρ ≠ 0 (Correlation coefficient is significant)

Decision: Reject H₀

Answer c)

In this case since magnitude of correlation coefficient is 0.7977 which is greater than 0.75 so effect size is large

Answer d) There was a significant negative relationship between family income and medical care expenditures.

Explanation:

In part b) we rejected null hypothesis. This means that correlation coefficient is significant. Also, since value of correlation coefficient is negative, we can say relationship between two variable is negative.