91Ó°ÊÓ

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

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

$H_0:\mathrm{μ}=200$

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.

$H_1:\mathrm{μ}≠200$

$\mathrm{μ}$is the mean response time of servers in Europe

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 $\mathrm{μ}$is appropriate.

$95\%$ confidence interval:

$(158.22,189.64)$

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.

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.