Question

The drying time of a type of paint under specified test conditions is known to be normally distributed with mean value 75 min and standard deviation 9 min. Chemists have proposed a new additive designed to decrease average drying time. It is believed that drying times with this additive will remain normally distributed with σ = 9 min. Because of the expense associated with the additive, evidence should strongly suggest an improvement in average drying time before such a conclusion is adopted. Construct a suitable hypothesis test to check the feasibility of using the paint with the additive based on a sample of 25 painted specimens which resulted in an average drying time of 72.3 min.

 Use a Fixed Significance level-test and state your conclusions.

 Repeat the test with a P-Value approach test and state your conculsions.

 Construct an upper bound confidence interval and test the hypothesis based on it.

Answer 1

SOLUTION- WE SHALL USE MINITAB 16 FOR THE CALCULATIONS:

LET DENOTE THE AVERAGE DRYING TIME FOR THE OLD ADDITIVE AND DENOTE THE AVERAGE DRYING TIME FOR THE NEW ADDITIVE.

AS THE NEW ADDITIVE WAS DESIGNED TO REDUCE THE AVERAGE DRYING TIME, THE HYPOTHESIS WHICH COULD BE FRAMED AND TESTED HERE IS-

WE PERFORM A TWO SAMPLE-T TEST IN ORDER TO EXAMINE THE ABOVE STATED HYPOTHESIS-

STEPS- STAT> BASIC STATISTICS> TWO SAMPLE-T> ENTER THE SUMMARIZED DATA(SAMPLE SIZE,MEAN AND S.D)> UNDER 'OPTIONS', SET THE CONFIDENCE LEVEL 95.0( WE ASSUME 5% LEVEL OF SIGNIFICANCE), AND ALTERNATE AS 'LESS THAN'> CLICK OK.

1.) WE OBTAIN THE VALUE OF TEST STATISTIC AS, T= 1.06

NOW FOR A TWO SAMPLE-T TEST,UNDER EQUAL VARIANCE ASSUMPTION, D.F= 25+25-2 = 48

CRITICAL VALUE=

AS T-VALUE< CRITICAL VALUE, WE REJECT OUR NULL HYPOTHESIS AND CONCLUDE THAT THE NEW ADDITIVE REDUCES THE AVERAGE DRYING TIME

2.) THE OBSERVED P-VALUE IN THIS CASE IS 0.853

AS P-VALUE>LEVEL OF SIGNIFICANCE(0.05), WE ACCEPT THE NULL HYPOTHESIS AND CONCLUDE THAT THE AVERAGE DRYING TIME FOR THE NEW ADDITIVE IS NEARLY THE SAME AS THE PREVIOUS ONE AND THERE IS NO SIGNIFICANT DECREASE IN THE DRYING TIME.

3.) THE UPPER 95% CONFIDENCE INTERVAL IS 6.97

ALSO, THE ESTIMATE OF THE DIFFERENCE OBTAINED IS 2.70

WE KNOW THAT A 95% CONFIDENCE INTERVAL FOR A LEFT TAILED TEST IS  

AS THE CONFIDENCE INTERVAL CONTAINS THE ESTIMATE OF NULL HYPOTHESIS, THE RESULTS OF THE TESTING ARE NOT STATISTICALLY SIGNIFICANT.

**REMARK**- THE NECESSARY EXPLAINATIONS HAVE BEEN GIVEN ABOVE. IN CASE OF DOUBT, COMMENT BELOW. AND PLEASE LIKE, IF YOU LIKED THE SOLUTION.