An investigator believes that families in Virginia have more

An investigator believes that families in Virginia have more children on average than the average number of children in the population of the United States. From census data the investigator knows that the mean number of children in the U.S. is 2.1 children. She draws a sample of seven families from the state of Virginia and obtains the number of children for each family. The data from this sample are presented below. Write the null and alternative hypotheses for testing the investigator’s prediction. Please use SPSS to test the null hypothesis stated above. Please provide a sentence in APA format for a results section that states the conclusion the investigator is entitled to draw.

Scores for the seven families: 3, 6, 2, 0, 4, 6, 7

Solution

let X be the random variable denoting the number of children to families of virginia

let mu be the average number of children to families of virginia.

assume that X~N(mu,sigma²)

the investigator believes that families in Virginia have more children on average than the average number of children in the population of the United States. From census data the investigator knows that the mean number of children in the U.S. is 2.1 children.

so here the null hypothesis is H0:mu=2.1    vs alternative hypothesis H1:mu>2.1

to test this he takes a random sample of size=n=7 from X

the values are 3,6,2,0,4,6,7

let Xbar be the sample mean. then Xbar~N(mu,sigma²/7)

or (Xbar-mu)*sqrt(7)/sigma~N(0,1)

now under H0 mu=2.1

so (Xbar-2.1)*sqrt(7)/sigma~N(0,1) under H0

but here sigma is unknown. it is replaced by s=sample standard deviation

then we have (Xbar-2.1)*sqrt(7)/s~t_7-1=6 under H0

so the test statistic is T=(Xbar-2.1)*sqrt(7)/s

we reject H0 at alpha=0.05 level of significance iff t>t_0.05;6

where t is the observed value of T and t_0.05;6 is the upper 0.05 point of t distribution with df 6

now Xbar=4 and s=2.517

so t=(4-2.1)*sqrt(7)/2.517=1.997

and t_0.05;6=1.94318

so t>t_0.05;6

hence H0 is rejected.

hence the conclusion is

BASED ON THE GIVEN DATA AT HAND AND AT 5% LEVEL OF SIGNIFICANCE H0 IS REJECTED AND IT IS CONCLUDED THAT THE FAMILIES IN VIRGINIA HAVE MORE CHILDREN ON AVERAGE THAN THE AVERAGE NUMBER OF CHILDREN IN THE POPULATION OF THE U.S.