Question

A product developer is interested in reducing the drying time of a primer paint. Two formulations of the paint are tested: Formulation-1 is the standard chemistry and formulation-2 is a new drying ingredient that should reduce the drying time. From experience, it is known that the standard deviation drying tine is 8 min and it should not be affected by addition of new ingredient. 10 specimens are painted with each formulation and data is tabulated below:
Parameter Formulation-2 Formulation- 1
Average 121 min 112 min
Sample size 10 10

Required:
a. By hypothesis testing method, check if the addition of new ingredient reduces the drying time. (Write the hypothesis, test statistics, critical region decision). Use alpha = 0.05.
b. What is the P-value of your test?
c. Draw P-value and a value in a standard normal distribution curve.

Answer 1

Answer:

the answer is in the explanation

Step-by-step explanation:

we are given

          sample size $n_{1} = n_{2}$ = 10  for each formulation

          mean  $\bar{x} _{1}$ (formulation 1) = 121

          mean $\bar{x} _{2}$  (formulation 2) = 112

         s (standard deviation )  =  8 mins for  each case

        null hypothesis        $H_{0}$        μ2 = μ1        (both have average same time)

 alternative hypothesis   $H_{1}$         μ2 < μ1

          under $H_{0}$  the test statistics is

                                $t = \frac{\bar{x}_{2} - \bar{x}_{1} }{s\sqrt{\frac{1}{n_{1} }+ \frac{1}{n_{1}} } }$

                               $t = \frac{112 - 121 }{8\sqrt{\frac{1}{10 }+ \frac{1}{10} } }$

                                t = -2.5

                        ItI = 2.5

            The P-value at  ItI = 2.5       at $\alpha$ = 0.05    μ = 0.010699

check the remaining solution and diagram in the attached image