Question

Fast connection? How long does it take for a chunk of information to travel

from one server to another and back on the Internet? According to the site

internettrafficreport.com, the average response time is 200 milliseconds (about one-fifth of a second). Researchers wonder if this claim is true, so they collect data on response times (in milliseconds) for a random sample of 14 servers in Europe. A graph of the data reveals no strong skewness or outliers.

a. State an appropriate pair of hypotheses for a significance test in this setting. Be sure to define the parameter of interest.

b. Check conditions for performing the test in part (a).

c. The 95% confidence interval for the mean response time is 158.22 to 189.64

milliseconds. Based on this interval, what conclusion would you make for a test of the hypotheses in part (a) at the 5% significance level?

d. Do we have convincing evidence that the mean response time of servers in the United States is different from 200 milliseconds? Justify your answer.

Short answer

Part (a)$$\begin{gathered} H_0:\mu =200 \\ {H}_1:\mu \ne 200 \end{gathered}$$

Part (b) All conditions are satisfied.

Part (c) There is enough convincing proof that the mean response time for serves in Europe is different from $200$ milliseconds.

Part (d) No.

Step-by-step solution

Part (a) Step 1: Given information

The claim is that the mean is $200$ milliseconds.

Part (a) Step 2: Explanation

The null hypothesis declares that the population value is the same as the claim value:

$H_0:\mu =200$

Either the null hypothesis or the alternative hypothesis is the assertion. The null hypothesis asserts that the population means is the same as the value stated in the claim. If the claim is the null hypothesis, the alternative hypothesis statement is the polar opposite of the claim.

$H_1:\mu \ne 200$

$\mu$is the mean response time of servers in Europe

Part (b) Step 1: Explanation

The three conditions are Random, independent, and Normal/ Large sample.

Random: Because the sample is a random sample, I'm satisfied.

Independent: because the sample of $14$ servers represents less than 10% of the total number of servers

Normal/Large sample: Because a graph of the data displays no severe skewness or outliers, I'm satisfied.

Because all of the prerequisites are met, a hypothesis test for the population means $\mu$is appropriate.

Part (c) Step 1: Given information

$95\%$ confidence interval:

$(158.22,189.64)$

Part (c) Step 2: Explanation

A hypothesis test with a significance level of $100\%-95\%=5\%$ equates to a $95$ percent confidence interval.

It is noted that the confidence interval does not include $200$ implying that the mean response time for servers in Europe is unlikely to be $200$ milliseconds, and so there is sufficient persuasive evidence that the mean response time for servers in Europe is not $200$ milliseconds.

Part (d) Step 1: Explanation

Because this data pertains to servers in Europe, it is impossible to draw any conclusions regarding servers in the United States, as their response times may differ from those in Europe. This indicates there's no compelling evidence that the average response time of servers in the United States is less than $200$ milliseconds.