# Confidence Interval, p-value and Critical Region

#### amr21

##### New member
Hello, I have been struggling with this question for a few days now would appreciate being walked through it!

Data are collected on the wingspans of adult robins. For N=20 birds, the sample mean and variance are given by $$\displaystyle \overline{x}$$=9.5cm and $$\displaystyle s^{2}=2.6^{2}cm^{2}$$

a) If we assume that the true population variance, $$\displaystyle \sigma^{2}$$, is known to be $$\displaystyle 2.6^{2}cm^{2}$$ (i.e. using a Z-test), construct a 95% confidence interval for the population mean.

b) What is the p-value for testing the null Hypothesis $$\displaystyle {H}_{0}:\mu=10cm$$ against $$\displaystyle {H}_{A}:\mu<10cm$$

c) What is the p-value for testing $$\displaystyle {H}_{0}:\mu=8.9cm$$ against $$\displaystyle {H}_{1}:\mu\ne8.9cm$$

d) For the hypothesis of part c what is the critical region for $$\displaystyle \alpha=0.01$$?

- I am unsure if I am doing part a correctly, I think it is $$\displaystyle \overline{x}~N(9.5, 0.581)$$, where 0.581 is $$\displaystyle \sqrt{\frac{2.6^{2}}{20}}$$ and I think the confidence interval is calculated using $$\displaystyle \overline{x}\pm1.96\frac{\sigma}{\sqrt(20)}$$, is this correct?

Thanks for any help with the next parts, pretty new to stats so I think the wording is what's confusing me!