# Learning about a population mean

1. Assuming $H_0$ is true, what is the distribution of the test statistic? t(29)

2. Assuming $H_0$ is true, what is the expected value of the test statistic? 87.70

3. The sample mean of 87.7 was ___ standard errors below the mean of 90. 1.615

Can someone verify that my work is correct? In particular, I'm not entirely sure about the second question. Will the test statistic actually be 0 in this case (as it wouldn't deviate from the mean)?

• For #2, it will be enlightening to state explicitly what $H_0$ asserts and then sit back and re-read the question. – whuber Mar 14 '13 at 22:14

## 1 Answer

Assuming H0 is true, what is the distribution of the test statistic? t(29)

No doubt this is the answer that was desired. However, it's strictly speaking incorrect -- without additional assumptions, the only possibly answer is "We can't tell".

Assuming H0 is true, what is the expected value of the test statistic? 87.70

This is not correct. You have given the sample mean of the random variable.

You are being asked for the population mean of a t(29) distribution; your later surmise at the end is thinking more along the right kind of lines, but you display some confusion here that should lead you to ask about the difference, or at least to ask about something.

The sample mean of 87.7 was _ standard errors below the mean of 90. 1.615

The question could be slightly clearer but I believe you have it right.

• For the second question, should the answer actually be 90, then? As then there is no deviation from the mean being tested in the hyptheses? – Bob John Mar 14 '13 at 22:31
• No. What is the thing you're being asked for the mean of? – Glen_b -Reinstate Monica Mar 14 '13 at 22:34
• The hypotheses are H0: u = 90 vs Ha: u < 90 – Bob John Mar 14 '13 at 22:37
• Thanks, that should go in your question, but I didn't ask for the hypothesis. What is the object that question two asks you to give the mean of? You seem to be just randomly grabbing quantities out of the question in the hopes that you have the right answer. You need to stop and actually think about what it is saying. – Glen_b -Reinstate Monica Mar 14 '13 at 22:48
• The advantage to you of me not just telling you the answer is that you get to discover you already had all the information (probably in your head). You actually know more than you realize. So not only are you now right, but you discover you already understood something that maybe it didn't seem like you did. You seem to be doing okay. – Glen_b -Reinstate Monica Mar 15 '13 at 0:57