Confidence Interval and P-value method

Here is the question and it's actually 2 parts.

Use the confidence interval method: For a sample of 25 items from a normally distributed population for which the standard deviation is o=10, the sample mean is 235. At the 0.05 level of significance, test Ho: u is less than or equal to 220 versus u > 220. Determine the 95% confidence interval. Is 220 a plausible value?

the second part:

Now use the p-value method to determine is 220 is plausible value. State the p-value and your conclusions. Use an alpha= 0.05.

Now, I from what I calculated, I came up with (231.08-238.92) as the 95% confidence level, which makes 220 not a plausible value. However, I am completely lost as where to go with the p-value. I have been working on this problem for hours and have gotten nowhere. Please help!


TS Contributor
you need to find the probability of observing a X=5*(220-235)/10

> pnorm(5*(220-235)/10)
[1] 3.190892e-14
> 235+c(-1,1)*1.96*10/5
[1] 231.08 238.92


TS Contributor
First of all, what's a p-value

In statistical hypothesis testing, the p-value is the probability of obtaining a value of the test statistic at least as extreme as the one that was actually observed, given that the null hypothesis is true. The fact that p-values are based on this assumption is crucial to their correct interpretation.

More technically, a p-value of an experiment is a random variable defined over the sample space of the experiment such that its distribution under the null hypothesis is uniform on the interval [0,1]. Many p-values can be defined for the same experiment.
from Wikipedia

Using a normal distribution this goes for your case as far as computing

Pr(X>5*(220-235)/10)=1-Φ(5*(220-235)/10), where Φ the distribution function of the standard normal (N(0,1))

This computation yields 3.190892e-14=0 which rejects the null hypothesis