Question

Previous research suggests that musicians process music in the same cortical regions in

which adolescents process algebra. So, a researcher wondered if receiving music

instruction while learning algebra would improve students’ grasp of algebra. On the other

hand, the researcher also thought it was possible that the addition to students’ already

crowded work and course schedules might give them less time to study and complete

algebra homework. So, to examine the connections between musical instruction and

algebra skills, the researcher collected data on a sample of 6,026 ninth grade students in

Maryland who had completed introductory algebra. Of these students, 3,239 received

formal music instruction (either choral or instrumental) during all three years of middle

school, while the remaining students had not. Of those receiving formal music instruction,

2,818 received a passing grade on the Maryland Algebra/Data Analysis High School

Assessment (HAS). In contrast, 2,091 of the 2,787 students who did not receive musical

instruction received a passing grade on the HAS. Is there a difference in the proportion

passing the HAS between students with and without formal musical instruction? To answer

this question, complete a two‐sample Z test with the following steps

1. State hypotheses in sentences and notation.
2. Calculate the pooled sample proportion.
3. Compute the Z test statistic.
4. Estimate the p‐value that corresponds to the test statistic.
5. State your conclusion in reference to the null hypothesis. (Use α = .01)
6. Interpret your result in reference to the alternative hypothesis.

Answer 1

p₁ : Proportion passing HAS students with formal musical instruction

p₂ : Proportion passing HAS students without formal musical instruction

Null hypothesis : H_o : There is no difference in the proportion passing the HAS between students with and without formal musical instruction

Alternate hypothesis :There is difference in the proportion passing the HAS between students with and without formal musical instruction   0

Null hypothesis : H_o : p₁ - p₂ =0

Alternate Hypothesis : H_a : p₁-p₂

Number of students who have received formal music instruction :n₁ =3239

Number of students who have received formal music instruction passed HAS :x₁ =2818

Number of students who have not received formal music instruction :n₂ =2787

Number of students who have received formal music instruction passed HAS :x₂ =2091

pooled sample proportion = 0.8146

Z test Statistic = 11.9214

For two tailed test:

P(Z<-11.9214) =0.0000;

p-value = 2 x P(Z<-11.9214) =0.0000;

p-value =0.0000

As P-Value i.e. is less than Level of significance i.e (P-value:0 < 0.01:Level of significance); Reject Null Hypothesis

There is sufficient evidence to conclude that there is a difference in the proportion passing the HAS between students with and without formal musical instruction