Miller - Chapter 5 -Hypothesis Testing and Confidence Intervals, pg 91 #2

[Dr. Jayanthi Sankaran]

Hi Shakti,

In the above question #2

A random sample of 36 observations drawn from a normal population returns a sample mean of 18.0 with sample variance of 16.0. Our hypothesis is: the population mean is 15.0 with population variance of 10.0. Which are nearest, respectively, to the test statistic of the sample mean and sample variance (given the hypothesized values, naturally)?

Answer: (b) t-stat of 4.5 and chi-square stat of 56.0

I get the above answers easily. For the chi-square test statistic of 56.0 the critical value turns out to be 46.059 at 95% confidence level and so we reject the null hypothesis.
However, we are unable to reject the null hypothesis at 99% confidence because the critical value is 57.342.
All this is okay - my question is, how do we determine the p-value at 35 degrees of freedom and Probabilities of 95% and 99%? Although, the question does not ask this, I am curious to know and learn. Thanks!

Jayanthi

 

[ShaktiRathore]

Hi
Jayanthi p value gives you the max. Confidence level CL at which u can safely reject the null. Find the corresponding CL say x% for chi test stat 56 here and 35 df in the chi sq table ,then p value is 1- x%. For egin above chi square test for sample variance chi-test stat=56 the area under the chi square dist to the right of this test stat is the p value so CL=1-p value which is the max Cl at which we can reject the null,at Cls at this or below we can safely reject the null. If the calc chiteststat was 46.059 then we could have safely reject null at 95% Cl and p valu would be 1-.95=.05 or 5%.A lower p value favors rejectiin of the null,p value is also calles exact significance level.
Thanks

 

[Dr. Jayanthi Sankaran]

Hi Shakti,

Thanks for your answer. I understand that p value gives you the maximum confidence level at which you can reject the null. For the chi-square test statistic of 56.0 the critical value is 46.059 at 95% confidence level, and the p value is 1-.95 = 5%. and so we reject the null hypothesis. At 97.5% confidence level, p value = 1-.975 = .025, the critical value = 53.203. So we reject the null again. It is only at 99% confidence that we are unable to reject the null where the p value = .01.
You have said a lower p value favors rejection of the null. However, at p value of 0.01 we are unable to reject the null because 56.0 < 57.342 (critical chi-square value at 99% confidence, 35 degrees of freedom). Can you please clarify?

Thanks!
Jayanthi

 

[ShaktiRathore]

Hi jayanthi
I have said it for a fixed CL. I mean at 99% CL as the p value decrease the chances of rejection increases, the CL are fixed its not varying. Likeeise at 95% CL if you decrease p value the chnces of rej increase. At 99% Cl if calc chistat were s.t. Greater than 57.32 than corresponding p valuu would be smaller as compared to when chistat were smaller than 57.32. p value of .01 is max required to reject nul at 99% CL as p value decreases to .05 ans so on the null is rejected while if p value increase to .10 and so on the chance of rejection nullifies. Therefore as p value decrease at ai given Cl and df the chance of rej increase.i think it helps u undesrtood,
Tha

 

[Dr. Jayanthi Sankaran]

I am sorry, I still don't understand, Shakti. For 99% confidence level, the p value = 1 - .99 = .01 - we cannot reject null because chi-square statistic 56<57.342. As p value declines from 0.01 to 0.001 the critical value increases from 57.342. With the test statistic of 56, we are still unable to reject the null. Also, I thought that from David's notes that confidence intervals, always vary, because of varying samples. It is only the population mean that is static, not the confidence intervals - such that in the Hypothesis testing we state: that confidence level which contains the mean. I must be missing something.

Thanks!
Jayanthi

 

[brian.field]

I don't really answer your question, but I thought I would add two more cents....

CI’s are always two-sided. If random samples are obtained over and over again, then the unknown population parameter βj would lie in the interval for (1-α) % of the samples. We can’t know whether the parameter is actually in the interval constructed from the actual sample used to produce the CI. We hope that we have obtained a sample that is one of the 95% of samples for which the interval would contain the parameter. As such, the CIs do indeed change with each sample. Also, if heteroscedasticity is present, then the standard error is not a reliable estimate of the standard deviation of βj and the CI constructed using these assumptions will not truly reflect the degree of confidence chosen.

 

[Dr. Jayanthi Sankaran]

Thanks, Brian - for elaborating on what I had said on confidence intervals being varying, while the population mean parameter remains static.

Jayanthi

 

[Dr. Jayanthi Sankaran]

Hi Shakti and Brian,

I think I am beginning to understand what Shakti, said earlier. In the given problem, chi-square test statistic = 56.0. Critical chi-square value (35 df, pr .01) ie. 99% confidence at the p value = 1.357% is 57.342. Hence we cannot reject the null hypothesis since 56 < 57.342. In other words the result is not significant at p < 0.01 - in other words 98.64% is the area under the curve (1 - 0.01357) to the left of the critical chi-square value. However, that is not entirely true because the chi-square test statistic falls between 97.5% and 99% confidence level corresponding to critical values in the range of 53.203 and 57.342.

Critical chi-square value (35 df, pr 0.05) ie 95% confidence at the p value = 1.357% is 49.802. Hence we reject the null hypothesis since 56 > 49.802. In other words the result is significant at p < 0.05. As the p value decreases at the 95% confidence, keeping it constant, the probability of rejecting the null hypothesis increases.

Critical chi-square value (35 df, pr.10) ie 90% confidence at the p value = 1.357% is 46.059. Hence we reject the null hypothesis since 56>46.05. In other words, the result is significant at p < 0.10. As p value decreases at fixed confidence 90%, the probability of rejecting the null hypothesis increases.

I just couldn't let it go!

Thanks
Jayanthi

 

[ShaktiRathore]

Jayan

I am sorry, I still don't understand, Shakti. For 99% confidence level, the p value = 1 - .99 = .01 - we cannot reject null because chi-square statistic 56<57.342. As p value declines from 0.01 to 0.001 the critical value increases from 57.342. With the test statistic of 56, we are still unable to reject the null. Also, I thought that from David's notes that confidence intervals, always vary, because of varying samples. It is only the population mean that is static, not the confidence intervals - such that in the Hypothesis testing we state: that confidence level which contains the mean. I must be missing something.

Thanks!
Jayanthi

Hi
Yes p value is associated with the test stat not CL 99%. If chistat=56 then we find p value associated and compare it with .01 if p value<.01 then we reject null otherwise if pval>.01 fail to rej null. In this case of 56 yes p val>.01 so we fail to reject null. If p value is declining then chitest stat is increasing from 56 such that at one level its >critical value 57.32 and we reject the null. Thus decreasing p value increase rejection chances.
Thanks