Question

Use the given statistics to complete parts​ (a) and​ (b). Assume that the populations are normally distributed. ​(a) Test whether mu 1greater thanmu 2 at the alphaequals0.10 level of significance for the given sample data. ​(b) Construct a 99​% confidence interval about mu 1minusmu 2. Population 1 Population 2 n 25 19 x overbar 46.9 43.1 s 6.9 12.3 ​(a) Identify the null and alternative hypotheses for this test. A. Upper H 0​: mu 1less thanmu 2 Upper H 1​: mu 1equalsmu 2 B. Upper H 0​: mu 1equalsmu 2 Upper H 1​: mu 1less thanmu 2 C. Upper H 0​: mu 1equalsmu 2 Upper H 1​: mu 1not equalsmu 2 D. Upper H 0​: mu 1greater thanmu 2 Upper H 1​: mu 1equalsmu 2 E. Upper H 0​: mu 1equalsmu 2 Upper H 1​: mu 1greater thanmu 2 F. Upper H 0​: mu 1not equalsmu 2 Upper H 1​: mu 1equalsmu 2

Answer 1

The given problem is to test whether , thus we use two independent sample t test with unequal variance to test this hypothesis.

GIVEN:

Sample size from population 1

Sample size from population 2

Sample mean from population 1

Sample mean from population 2

Sample standard deviation from population 1

Sample standard deviation from population 2

(a) HYPOTHESIS:

(That is, the mean of population 1 is less than or equal to the mean of population 2.)

(That is, the mean of population 1 is greater than the mean of population 2.)

LEVEL OF SIGNIFICANCE:

TEST STATISTIC:

which follows t distribution with degrees of freedom given by,

and the hypothesized value

DEGREES OF FREEDOM:

  

  

CRITICAL VALUE:

The right tailed (since ) t critical value with 27 degrees of freedom at is .

CALCULATION:

DECISION RULE:

INFERENCE:

Since the calculated t statistic value (1.21) is less than the t critical value (1.314), we fail to reject null hypothesis and conclude that the mean of population 1 is less than or equal to the mean of population 2.

(b) 99% CONFIDENCE INTERVAL FOR DIFFERENCE IN TWO POPULATION MEANS:

The formula for 99% confidence interval for two population mean is,

where is the t critical value with 99% confidence level.

The right tailed t critical value at 99% confidence level with 27 degrees of freedom is .

The 99% confidence interval for difference in two population means is,

The 99% confidence interval for difference in two population means is .