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

hypothesis testingself-study

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

Best Answer

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.

Related Solutions

Solved – Simple multiple choice for test statistic and significance level

1) It depends on the test statistic! You can sensibly get positive or negative test statistics, but the two I'd most likely use (two sample proportions test of $\pi_2-\pi_1$ OR chi-square) would both be positive.

2) There's not enough information - we don't have sample sizes!

[Well, in fact we sort of can sometimes figure out a difference must be significant even without the sample sizes, just based on the counts that can give us the observed proportions:

i) What's the smallest sample size in which we could observe 45%? Assuming it's a rounded fraction, it looks like 5/11

ii) what's the smallest sample size in which we could observe 53%? That looks like 8/15 is the smallest possible.

So if it's significant at the lowest possible sample size, and the next few 45%-vs-53% sample-sizes up (to allow for the fact that the actual difference might get a little smaller and more than undo the gain in smaller standard errors), it will be significant at larger samples. But in this case it turns out it's not significant for 5/11 vs 8/15:

> prop.test(c(5,8),c(11,15))

2-sample test for equality of proportions with
continuity correction

data:  c(5, 8) out of c(11, 15) 
X-squared = 0, df = 1, p-value = 1
alternative hypothesis: two.sided 
95 percent confidence interval:
 -0.5452923  0.3877165 
sample estimates:
   prop 1    prop 2 
0.4545455 0.5333333

So that's no help - since it's not significant at the smallest sample sizes, we still can't tell for sure. I assume you have sample size information elsewhere that you didn't notice.]

Solved – Choice of null and alternative hypothesis

As explained in this answer What follows if we fail to reject the null hypothesis? one can only find 'statistical evidence' for $H_a$ if the sample is such that $H_0$ is rejected. If $H_0$ can not be rejected then the only conclusion one can draw is that $H_0$ does not lead to a statistical contradiction.

In other words if one can not reject $H_0$ that does not imply that $H_0$ is true, this is the reason why we say that it is ''accepted'' (which is not the same as statistically proven).

In your example, the first hypothesis test $H_0: \mu \le 1.5$ versus $H_a: \mu > 1.5$ you find a p-value of 0.305 which means that your sample does not provide ''evidence'' for $H_a$. This does however not imply that you can conclude that $H_0$ is true (it is only not rejected or accepted, but there is no evidence for $H_0$.

In your second situation you test $H_0: \mu \le 1.5$ versus $H_a: \mu > 1.5$ you find a p-value of 0.695, so you have no evidence thet $H_a$ is true. The fact that you can not prove that $H_a$ holds does not imply that $H_0$ holds.

The only conclusion that you can draw from your example is that your sample does not provide evidence for $H_a: \mu < 1.5$ nor does it provide evidence for $H_a: \mu > 1.5$.

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