Question

# For each problem students will write out all steps of hypothesis testing including populations, hypotheses, cutoff...

For each problem students will write out all steps of hypothesis testing including populations, hypotheses, cutoff scores, and all relevant calculations.

A psychologist has developed a standardized test for measuring the vocabulary skills of 4-year-old children. The scores on the test form a normal distribution with μ = 60 and σ = 10. A researcher would like to use this test to investigate the idea that children who grow up with no siblings develop vocabulary skills at a different rate than children in large families. A sample of n = 25 children is obtained, and the mean test score for this sample is 63. On the basis of this sample, can the researcher conclude that vocabulary skills for children with no siblings are significantly different from those of the general population? Test at the .01 level of significance

To Test :-
H0 :- µ = 60
H1 :- µ ≠ 60

Test Statistic :-
Z = ( X̅ - µ ) / ( σ / √(n))
Z = ( 63 - 60 ) / ( 10 / √( 25 ))
Z = 1.5

Test Criteria :-
Reject null hypothesis if | Z | > Z( α/2 )
Critical value Z(α/2) = Z( 0.01 /2 ) = 2.576 ( from Z table )
| Z | > Z( α/2 ) = 1.5 < 2.576
Result :- Fail to reject null hypothesis

Decision based on P value
Reject null hypothesis if P value < α = 0.01 level of significance
P value = 2 * P ( Z > 1.5 ) = 2 * 1 - P ( Z < 1.5 )
P value = 0.1336
Since 0.1336 > 0.01 ,hence we reject null hypothesis
Result :- We fail to reject null hypothesis

There is insufficient evidence to support the claim that vocabulary skills for children with no siblings are significantly different from those of the general population.