Consider the following competing hypotheses and accompanying sample data drawn i

Question

Consider the following competing hypotheses and accompanying sample data drawn independently from normally distributed populations. Use Table 2.

H0: 1 2 0

HA: 1 2 < 0

X1 = 259       X2 = 267

s1 = 31         s2 = 19

n1 = 6            n2 = 6

a-1. Calculate the value of the test statistic under the assumption that the population variances are unknown but equal. (Negative values should be indicated by a minus sign. Round intermediate calculations to 4 decimal places and final answer to 2 decimal places.) Test statistic

a-2. Calculate the critical value at the 10% level of significance. (Negative value should be indicated by a minus sign. Round your answer to 3 decimal places.) Critical value a-3. Do you reject the null hypothesis at the 10% level? Yes, since the value of the test statistic is not less than the critical value. No, since the value of the test statistic is less than the critical value. No, since the value of the test statistic is not less than the critical value. Yes, since the value of the test statistic is less than the critical value.

b-1. Calculate the value of the test statistic under the assumption that the population variances are unknown and are not equal. (Negative values should be indicated by a minus sign. Round intermediate calculations to 4 decimal places and final answer to 2 decimal places.) Test statistic

b-2. Calculate the critical value at the 10% level of significance. (Negative value should be indicated by a minus sign. Round your answer to 3 decimal places.) Critical value

b-3. Do you reject the null hypothesis at the 10% level? Yes, since the value of the test statistic is not less than the critical value. No, since the value of the test statistic is not less than the critical value. Yes, since the value of the test statistic is less than the critical value. No, since the value of the test statistic is less than the critical value.

Explanation / Answer

Set Up Hypothesis
Null, Ho: u1 - u2 >=0
Alternative, H1: u1 - u2 < 0
Test Statistic
X (Mean)=259; Standard Deviation (s.d1)=31
Number(n1)=6
Y(Mean)=267; Standard Deviation(s.d2)=19
Number(n2)=6
Value Pooled variance S^2= (n1-1*s1^2 + n2-1*s2^2 )/(n1+n2-2)
S^2 = (5*961 + 5*361) / (12- 2 )
S^2 = 661
we use Test Statistic (t) = (X-Y)/Sqrt(S^2(1/n1+1/n2))
to=259-267/Sqrt((661( 1 /6+ 1/6 ))
to=-8/14.8436
to=-0.539
| to | =0.539
Critical Value
The Value of |t | with (n1+n2-2) i.e 10 d.f is 1.372
We got |to| = 0.539 & | t | = 1.372
Make Decision
Hence Value of |to | < | t | and Here we Do not Reject Ho
P-Value: Left Tail - Ha : ( P < -0.539 ) = 0.30086
Hence Value of P0.1 < 0.30086,Here We Do not Reject Ho

[ANSWERS]

a-1. to=-0.539
a-2. Critical Value = -1.372
b-1. to=-0.539
b-3 . Do not Reject Ho

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