Question

Problem 4) According to the most recent (2016) large-scale data analyzed by Kids Count Data Center, 45 percent of children in New Mexico live in single-parent households. Assume this is the baseline value. Sociologists in New Mexico want to see if the proportion of children in NM in single-parent households has changed in 2017. In 2017 they select a random sample of 835 children and note if they come from single-parent families: 415 did.

a) Consider just one experimental unit – that is, one child in the survey. What is the response variable? Categorical or quantitative?

b) A sociologist wants to conduct a significance test to decide if the percent of children from single-parent households has changed from the 2016 value of 45% in NM. Verify the conditions for using the normal approximation for the sample proportion.

c) Conduct the significance test. Keep two nonzero digits in your calculation for ?̂ and for the standard deviation. Sketch the distribution for ??̂ showing mean and area for P-value. State your conclusion in plain English in the context of the problem. Look at the statements for Problem 1 and use them as a guide.

Answer 1

The experiment is a survey about the single parent household.

4.

a. Considering the single experimental unit ie just a single child the response variable is Either Yes or No for the question whether the child is from a single-parent household? Therefore the response is categorical(one of the two categories Yes/No).

b. A sociologist wants to conduct a significance test to decide if the percent of children from single-parent households has changed from the 2016 value of 45% in NM. Verify the conditions for using the normal approximation for the sample proportion.

The conditions for using the Normal approximation:

1. p the probability of success should be closer to 0.5. Here our estimate is 415 out of 835 ie . It is close to 0.5 and this condition is satisfied.

2. The sample size n is large. We have n=835, which is large enough a sample and hence both the conditions are met and we can use a normal approximation.

c.Conduct the significance test

The population proportion of single parent children is which is 0.45(Previous year result)

The population proportion is not equal to .

Level of significance:

Test Statistic:

follows a Normal distribution with 0,1 as the parameters.

Here we have

  

  

  . The p-value can be obtained from the EXCEL as 2*NORM.S.DIST(-2.9,TRUE)=0.0037. Since p-value is less than <0.05, we reject the null hypothesis and hence, the population proportion is not equal to 0.45. The area for the p-value is given below in the shaded area.