Question

3. Let X and Y denote the tarsus lengths of male and female grackles, respectively. Assume that X is Nux, ox) and Y is Nuy, o). Given that nx = 25, i = 34.68, $ = 4.88; ny = 29, y = 32.55 and 5 = 5.81, test the null hypothesis Ho: Mix = My against H :MX > Hy with a =0.01. Hint: you need to justify whether it is reasonable to assume same variance for the two samples or not.

Answer 1

ANSWER:

Let X and Y denote the tarsus lengths of male and female grackles,Respectively.

To test the hypothesis first of all we shall check that population variances are equal or not. We shall check this by testing the following hypotheses

Ho: 0 = 0

H1:02 +0

To test the hypothesis we shall use F test statistic which is given by,

~ F (n1 - 1, m2 - 1)

We have
S2 = 4.88
and
S = 5.81

F= BO = 0.8399

The value of the test statistic is 0.8399.

The p-value for the test statistic is 0.6688.

Since, p-value for the test statistic is greater than the significance level of 0.01, therefore we shall be fail to reject H0 at 0.01 significance level.

Hence, population variances are equal.

The null and alternative hypotheses are as follows:

Ho: Mo = Hy

H, :μ2 μμ

To test the hypothesis we shall use two independent samples t-test. The test statistic is given as follows:

1 (ĩ – ) , –xt(n + n2 - 2) pooled kn + b)

Where,

2 Spooled =- (n1 - 1)5. + (n2 - 1) (n1 + n2-2)

We have ,
(-) = (34.68 -32.55) = 2.13.
  
$2 = 4.88

$=5.81,

ni = 25
and  
n2 = 29

Spooled = (25 - 1)4.88+ (29 - 1)5.81 25 + 29 - 2 = 5.3808

2.13 5.3808(+ == 3.3646 )

The value of the test statistic is 3.3646. The degrees of freedom is

(nu + n2 - 2) = 52.

Our test is right tailed test, therefore we shall obtain right tailed p-value for the test statistic. The right tailed p-value is given as follows:

p-value = P(T > t )

p-value = P(T > 3.3646) = 0.0007

The p-value for the test statistic is 0.0007. Given that significance level is α = 0.01.

(0.0007 < 0.01)

Since, p-value is less than the significance level of 0.01, therefore we shall reject H0 at 0.01 significance level.

Please rate the answer. Thank you.