The drying time of a certain type of paint under specified test conditions is kn

Question

The drying time of a certain 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 sigma = 9. Because of the expense associated with the additive, evidence should strongly suggest an improvement in average drying time before such a conclusion is adopted. Let mu denote the true average drying time when the additive is used. The appropriate hypotheses are H_0: mu = 75 versus H_a: mu

Explanation / Answer

Answer to part a)

To find the value of Type II error

We first of all find the Z critical value for alpha = 0.01 left tailed test

we get Z c= -2.33

.

Using the Z formula for n = 100 , we get

-2.33 = (x - 75) / (9/sqrt(100)

x = 72.903

.

Beta = P( z < (72.903-74) /(9/sqrt(100))

Beta = P(z < -1.22)

Beta = 0.1114

.

For n = 784

We use the Z formula

-2.33 = (x - 75) / (9/sqrt(784))

x = 74.25

.

Beta = P(z < (74.25-74)/9/sqrt(784))

Beta = P(z < 0.78)

Beta = 0.78165

.

For n = 2500

-2.33 = (x - 75) / (9/sqrt(2500))

x = 74.58

.

Beta = P(z < (74.58-74) / (9/sqrt(2500))

Beta = P( z < 3.22)

Beta = 0.9994

.

Answer to part b)

z = (74 -75) / (9/sqrt(2500))

z = -5.55556

.

The P value = 0.000

.

Since the P value is 0.000 it is always less than any value of alpha , thus the result is statistically significant and we caan reject the null hypothesis

The correct answer choice is Second option

.

Answer to part c)

With a sample size of 2500 , even a small difference is highglighted as a statistically significant difference, which is not correct. 75 to 74 can be counted as a practically insignificant or minor change. but just because the sample size is very large, this minor difference appears to be statistically significant whihc leads to a wrong conclusion. Thus it is advisable not to make use of such large sample

hence the correct answer choice is third statement

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