# Drawing conclusion from fixed significance level or p-value in a two-sample test

This was an example done in class, However I was sick

An experiment was performed to determine whether the average nicotine content of brand A cigarette exceeds that of brand B cigarette by 0.20 milligram. If 50 cigarettes of brand A had a sample mean of 2.61 milligrams whereas 40 brand B cigarettes had an average nicotine of 2.38 milligrams. The population standard deviations of the nicotine content for the two brands of cigarettes are known to be 0.12 and 0.14 for brand A and B, respectively.

(a) Based on a significance level of 5%, what can you conclude about the difference between the two brands of cigarettes?

(b) Base on a p−value, what can you conclude about the difference between the two brands of cigarettes?

My Attempt:

(a)

$H_{0} :\mu_{A}-\mu_{B} =0.2$

$H_{1} :\mu_{A}-\mu_{B} \ne 0.2$

Significance Level : $\alpha = 0.05$

Rejection Region : $|z| >1.96$

Test Statistic : $z = \frac{2.61-2.38 -0.2}{\sqrt{\frac{0.12^2}{50}+\frac{0.14^2}{40}}} =1.08$

Conclusion : Since $1.08 <1.96$ I fail to reject $H_{0}$ at 5%

I really need Help with B

• What did you try?
– ThiS
Jan 8 '13 at 13:09
• I added the homework tag; this reads very much like a HW problem. Jan 8 '13 at 13:24
• Welcome to the site, @Jason. Please don't remove the HW tag, even if this was an in-class question, & not technically homework. The tag doesn't exist just to label questions that come from someone's actual HW, but to identify any "routine question from a textbook, course, or test used for a class or self-study". Your Q does come from a course, & it seems you are using this for self-study, in a sense, now. You can read more about this here: should-we-tag-questions-that-smell-like-homework & on the FAQ. Jan 8 '13 at 14:06
• Is your issue with part (b) one of not knowing how to compute the p-value, or not knowing how p-values relate to the conclusion of a statistical test at a given significance level, or both? Jan 8 '13 at 15:38
• Your statement of the hypothesis is incorrect according to "n experiment was performed to determine whether the average nicotine content of brand A cigarette exceeds that of brand B cigarette by 0.20 mg" May 13 '14 at 18:19

## 2 Answers

The area of the standard normal curve corresponding to a z-score of 1.08 is 0.1251. Because this test is two-tailed, that figure is doubled to yield a probability of 0.2502 (25%) that the population means are the same.

• @Glen_b not knowing how p-values relate to the conclusion of a statistical test at a given significance level Jan 8 '13 at 17:20
• @Jason Given the definition of the p-value, if the p-value is larger than the significance level, what would that imply? What would it imply if it was less than or equal to the significance level? Try the first few sentences of en.wikipedia.org/wiki/P-value for further explanation. Jan 9 '13 at 2:31

The essential question seems to be "how to interpret a p-value". This is my favorite paper on the subject, which explains the evolution (and misuse) of null-hypothesis significance testing.

Edit Thanks @Ben Bolker, corrected. Well spotted.

The short answer is it's the probability of data equal to or more extreme than the observed values given the null hypothesis (which, of course, was stated before performing the experiment). In your case, the probability of the data in your experiment, given the means are in fact the same, is <5% (actually approx. <14% based on the distribution function for the standard normal distribution).

• this is false! it's the probability of data equal to or more extreme than the observed values, under the null. $P(D|H)$ is the likelihood, not the p-value. Jan 4 '14 at 15:14