Statistics 12

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A sample of 38 observations is selected from one population with a
population standard deviation of 3.4. The sample mean is 100.5.
A sample of 51 observations is selected from a second population
with a population standard deviation of 5.8. The sample mean is 98.8.
Conduct the following test of hypothesis using the 0.05 significance level.

H0 : μ0 = μ2

H1 : μ1 ≠ μ2

(a) This a ..... test.

(b) State the decision rule. (Negative amount should be indicated by a
minus sign. Round your answer to 2 decimal places.)

(c) Compute the value of the test statistic. (Round your answer to 2 decimal places.)

(d) What is your decision regarding H0?

(e) What is the p-value? (Round your answer to 4 decimal places.)

Solution: Assume that the 2 populations are normal distributions with
means μ1 and μ2 and a common population s.d. σ.

The 2 samples are randomly drawn and independent.

H0 : μ1 = μ2 against H1 : μ1 ≠ μ2

This is a two-tailed test. (ans a)

Given:

z = (xm1 – xm2 ) / sqrt (σ12/n1 + σ22/n2)

= (100.5 – 98.8) / sqrt ( 3.4 2/38 + 5.8 2/51)

= 1.7 / sqrt ( 0.304 + 0.659)

=1.7 / sqrt 0.963

= 1.7 / 0.981

= 1.7329

Value of the test statistic = 1.73 (ans c)

The decision rule is to reject H0 if z is outside the interval (-1.96 ,1.96 ).(ans b)

Since 1.73 < 1.96, so we cannot reject null hyp. H0. (ans d)

Z=1.73

The two-tailed P value equals 0.0836 (ans e)

By conventional criteria, this difference is considered to be not quite statistically significant.