Hi

@Harshit Chawla Maybe we should parse the math from the language, because it's Miller's question (I would not write a question like this,

*I do presume you understand this is Miller's question*) and the language is imprecise (and I probably over-explained it above

).

The math is simpler than the language. The

**observed sample mean is 45** and the null value is 40 so that, to agree with you, a candidate one-sided hypothesis is

- Null H(0): µ ≤ 40 --> Alternative H(1): µ > 40 (not a bad time to remind the reader that the null must contain the "'=")
- The test statistic is (45 - 40) / [29/sqrt(10)] = 0.54; this signifies that our observation of 45 is "merely" 0.54 standard standard deviations away from a null hypothesized value of 40.0
- The implied one-sided p-value = T.DIST.RT(0.54,9) = 30.12% and the two-sided (the more typical) = T.DIST.2T(0.54, D21-1) = 60.0%; large p-values expected because we are too close to the null to reject it!

If we really want to do this correctly,

**we should pause on these simple facts **because the question is flawed. A technically correct question would be:

*If the population mean truly is 40 or less (aka, conditional on a true one-sided null hypothesis), what is the probability of observing this sample mean of 45 or one that is more extreme? *Answer: the p-value of 30.12%, hence our inability to reject the null.
- A typical "shortcut paraphrase" of this
*correct* question, in turn, would be: *What is the probability the true mean is less thnn or equal to 40; i.e., what is the probability that the null is true?* ... that's what I mean by the question we'd normally expect in this setup. Statistics students will understand that the shortcut paraphrase itself is imprecise, but at the same time, I would point out that it's common! A proper understanding should pause here because ...

Miller commits a fallacy: because the p-value is 30%, he assumes the probability that the alternative is true is (1 - p ) or 70.0%. Visually (on the distribution), it's understandable, but the one-sided p value of 30.0% is an

*exact significance level* (aka, the probability of committing a Type I error which as part of its definition refers to an error that is

*conditional on a true null*). The p-value of 30.0%

**does not **imply a 70% probability that the alternative is true, which is what the questions asks, and hence commits a fallacy. (Re: Finally, can we be asked in the exam to compute our exact confidence? Doesn't that require us to compute the p value? Yes but we can infer p-value from the lookup table. Please search forum many many discussions and examples on this already). Okay, phew, I need to get to some other things now. I hope that is helpful,

(P.S. Although I do think your first impression reverses the basic distribution, so you may want to re-visit that. Our sample mean of 45 is to the right of the null hypothesized 40. The

**null of 40 is the median of the student's distribution**; and to the right of it, not very far, is the quantile of 45.0, and the area in the tail to its right is ~30.0%; aka, the one-tailed rejection region is a large 30.0% tail).

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