Question

1)   Explain why we never say “accept” the null hypothesis (but we rather say “fail to reject” the null hypothesis).  

2)   State the requirements that must be satisfied to test a hypothesis regarding a population mean with either the population standard deviation know or unknown.

3)   Explain two graphs/figures/plots that you can use to assess whether the sample data is normally distributed? Make sure to explain how to assess normality using these graphs; that is, what to look for in the graphs.

4) The procedures for testing a hypothesis regarding a population mean with known or unknown are robust. What do the procedures being robust mean?

5) Distinguish between using z-distribution to test hypotheses regarding the population mean and using a t-distribution to test hypotheses regarding the population mean?

Answer 1

1.

We consider a null hypothesis and an alternative hypothesis before performing a hypothesis test. Based on our alternative hypothesis if we find sufficient evidence against our null hypothesis, then we reject our null hypothesis. If we do not find sufficient evidence against our null hypothesis, we cannot reject it. This does not imply that the null hypothesis must be true. It may be such that for another suitable alternative hypothesis, we get sufficient evidence against our null hypothesis and draw proper conclusion. Thus we cannot accept null hypothesis as it is tested based on our alternative hypothesis (although we try our best to take appropriate alternative hypothesis for the test). Hence, we never say “accept the null hypothesis", but we rather say “fail to reject the null hypothesis".

2.

To test a hypothesis regarding a population mean with either the population standard deviation known or unknown, it is required that the population follows normal distribution (at least approximately normal distribution).

3.

To assess whether the sample data is normally distributed, we can use any of the following graphs.

• Histogram- If we observe that middle portion posses higher frequencies and two tails gradually decreases almost symmetrically, then we can say that the distribution is almost normally distributed.
• Bar plot- If we observe that middle portion posses higher frequencies and two tails gradually decreases almost symmetrically, then we can say that the distribution is almost normally distributed.

4.

The robust procedures for testing a hypothesis regarding a population mean with known or unknown population standard deviation are

• z-test
• t-test

5.

To test hypotheses regarding the population mean,

• we use z-test if population standard deviation is known
• we use t-test if population standard deviation is unknown