Question

ath & Music (Raw Data, Software Required):
There is a lot of interest in the relationship between studying music and studying math. We will look at some sample data that investigates this relationship. Below are the Math SAT scores from 8 students who studied music through high school and 11 students who did not. Test the claim that students who study music in high school have a higher average Math SAT score than those who do not. Test this claim at the 0.05 significance level.

	Studied Music	No Music
count	Math SAT Scores (x1)	Math SAT Scores (x2)
1	516	480
2	571	535
3	589	553
4	588	537
5	521	480
6	564	513
7	531	495
8	597	556
9		554
10		493
11		557

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u_Music = u_{No Music}
Alternative hypothesis: u_Music > u_{No Music}

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

Studied music		No music
Mean	559.625	Mean	523
Standard Error	11.51697	Standard Error	9.500239
Median	567.5	Median	535
Mode	-	Mode	480
Standard Deviation	32.57491	Standard Deviation	31.50873
Sample Variance	1061.125	Sample Variance	992.8
Kurtosis	-1.89149	Kurtosis	-1.82542
Skewness	-0.34246	Skewness	-0.28312
Range	81	Range	77
Minimum	516	Minimum	480
Maximum	597	Maximum	557
Sum	4477	Sum	5753
Count	8	Count	11

SE = sqrt[(s₁²/n₁) + (s₂²/n₂)]
SE = 14.9297
DF = 17

t = [ (x₁ - x₂) - d ] / SE

t = 2.45

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is thesize of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.

The observed difference in sample means produced a t statistic of 2.45.

Therefore, the P-value in this analysis is 0.012.

Interpret results. Since the P-value (0.012) is less than the significance level (0.05), hence we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that students who study music in high school have a higher average Math SAT score than those who do not.