Question

2. The response time of a distributed computer system is an important quality characteristic. The system manager wants to know whether the mean response time to a specific type of command exceeds 82 millisec. The computer systems are known to have a mean response time of 84.25 millisec and standard deviation of 10 millisec. The command is executed 25 times.
   1. Test the hypothesis that the mean response time is 82 millisec versus the alternative that the mean exceeds 82 millisec. Use a type I error of α 0.05.
      1. State the hypothesis

2. Calculate the test statistic
   3. State the conclusion

1. 1. 1. 2. Construct a 95% two-sided confidence interval on the mean response time.

Answer 1

The system manager wants to know whether the mean response time to a specific type of command exceeds 82 millisec.

Let X be the random variable which takes the response time of a distributed computer system.

Let's write the given information

n = sample size = 25

= sample mean = 84.25

s = sample standard deviation = 10

Level of significance = = 0.05

Here we want to know whether the mean response time to a specific type of command exceeds 82 millisec.

So the

the null hypothesis ( H₀ ) and the alternative hypothesis ( H_a ) are as follows:

From the alternative hypothesis the given test is right tailed test

Hypothesis testing

Here population standard deviation is not given and we use sample standard deviation(s) instead of population

standard deviation . Also sample size is not sufficiently large( >= 30) so we can assume the X follows normal distribution so we can use one sample t test.

Using minitab we get following result

The command is Stat>>>Basic Statistics >>1 sample t...

Select summary Statistics and fill the required information.

then click on Perform hypothesis test enter hypothesis mean (82)

Look the following image:

then click on Option select level of confidence = (1 - alpha)*100 = (1 - 0.05)*100 = 95.0

Alternative " greater than"

then click on Ok

We get the following output

From the above output

ii) t test statistic value = 1.13

p- value = 0.136

iii) Decision rule: 1) If p-value < level of significance (alpha) then we reject null hypothesis

2) If p-value > level of significance (alpha) then we fail to reject null hypothesis.

Here p value = 0.136 > 0.05 so we used first rule.

That is we fail to reject null hypothesis and accept alternative hypothesis.

At 5% level of significance we don't have sufficient evidence to conclude that the mean response time to a specific type of command exceeds 82 millisec .

b) Do the same process except select not equal instead of greater than in Alternative.

So we get the output as:

from the above output the 95% two-sided confidence interval on the mean response time is (80.12, 88.38)