Determine the hypotheses and justify them with a simple sent

Determine the hypotheses and justify them with a simple sentence describing what you’re testing.

Determine the correct distribution to use and briefly discuss why.

Generate your test statistic.

Determine the P-Value for comparison,

State your decision, whether you reject or fail to reject the null hypothesis and why.

Give a brief conclusion.

A statistics student heard her little brother claim that boys are smarte rthan girls in math. To test this, you take a random sample of math standardized test scores for boys and girls at his school with the following results:

Sample Size Math Score Sample Standard Deviation

Boys 35 86 4.38

Girls 32 88 5.23

Test whether boys have higher math scores on average that girls.

Solution

A statistics student heard her little brother claim that boys are smarte rthan girls in math. To test this, you take a random sample of math standardized test scores for boys and girls at his school with the following results:

Here we want to test the hypothesis that,

H0 : Boys and girls have equal score in maths.

H1 : boys have higher math scores on average that of girls (that is boys are smarter than girls)

Here we use t-test because sample information is given.

n1 = number of boys = 35

n2 = number of girls = 32

s1 = sample standard deviation of boys = 4.38

s2 = sample standard deviation of girls = 5.23

X1bar = mean math score of boys = 86

X2bar = mean math score of girls = 88

For testing first we want to check whether variances are equal or not.

The hypothesis for the test is,

H0 : variances are equal

H1 : variances are not equal

The test statistic is,

F = larger variance / smaller variance

= 5.23^2 / 4.38^2 = 1.4258

P-value -

EXCEL syntax : =finv(x , d.f.1 ,d.f.2)

x is the F-test statistic value

d.f.1 = n2 - 1 = 32 - 1 = 31

d.f.2 = n1 - 1 = 35 - 1 = 34

P-value = 0.1568

alpha = 0.05

P-value > alpha

fail to reject H0 at 5% level of significance.

Conclusion : Variances are equal.

So we use pooled variance here.

Pooled variance (S) =sqrt [ (n1 - 1) * s1^2 + (n2 - 1) * s2^2 / (n1+n2-2) ]

= sqrt [ 34 * 4.38^2 + 31 * 5.23^2 / 65 ]

S = 4.8042

The test statistic for testing two means is,

t = (X1bar - X2bar) / [S * sqrt(1/n1 + 1/n2) ]

t = (86 - 88) / [ 4.8042 * sqrt(1 / 35 + 1 / 32) ]

t = -2 / 1.1750

t = -1.7021

P-value :

EXCEL syntax : =tdist(x , d.f. , tails)

x is the test statistic value

d.f. = n1 +n2 - 2 = 35 + 32 - 2 =65

tails = 1

P-value = 0.047

P-value < alpha ( 0.05)

Reject H0 at 5% level of significance.

Conclusion : boys are smarte rthan girls in math.