Confidence Interval, p-value and Critical Region

[amr21]

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!

 

[Valkarie]

- For part b, the p-value is calculated using a one-tailed test. The null hypothesis is that the mean is 10 cm and the alternative hypothesis is that the mean is less than 10 cm. The formula for the p-value is P(Z<test statistic) = P(Z<[(\overline{x}-\mu_0)/\frac{\sigma}{\sqrt{N}}]) = P(Z<[(9.5-10)/\frac{2.6}{\sqrt{20}}]) = 0.039. - For part c, the p-value is calculated using a two-tailed test. The null hypothesis is that the mean is 8.9 cm and the alternative hypothesis is that the mean is not equal to 8.9 cm. The formula for the p-value is P(|Z|>test statistic) = P(|Z|>[(\overline{x}-\mu_0)/\frac{\sigma}{\sqrt{N}}]) = P(|Z|>[(9.5-8.9)/\frac{2.6}{\sqrt{20}}]) = 0.084. - For part d, the critical region for alpha=0.01 is the region of the z-distribution where z is greater than 2.575 or less than -2.575. This means that if the test statistic is greater than 2.575 or less than -2.575, then the null hypothesis should be rejected.