Business Finance Homework Help

Requirements:

For this homework, you need to work in groups of 3 or less.

For the questions where you do not need to use R, you can either type your answers, or you can solve them on paper, and upload a scanned document on Canvas.

• There will be one submission per group.
• Questions
• Question 1: (60 points) Consider a random sample of size nn from a Normal distribution: Xi?N(?,?2)Xi?N(?,?2).

If it is known that ?2=9?2=9, find a 90% confidence interval for ?? based on the estimate x¯=19.3x¯=19.3 and n=16n=16

Based on the information in part (a), find a one-sided 90% lower confidence limit for ??, also find a one-sided 90% upper confidence limit for ??.

For a confidence interval of the from (x¯?z(1??/2)?/n???,x¯+z(1??/2)?/n???)(x¯?z(1??/2)?/n,x¯+z(1??/2)?/n), derive a formula for the sample size required to obtain an interval of specific length ??. If ?2=9?2=9, what nn is needed to have a 90% confidence interval of length (?)(?) equal to 2?

Suppose now that ?2?2 is unknown. Find a 90% confidence interval for ?? when x¯=19.3x¯=19.3, s2=10.24s2=10.24 and n=16n=16.

Based on the data in part (d), find a 99% confidence interval for ?2?2.

Question 2: (40 points) Consider a random sample of size 2020 from a normal distribution: Xi?N(?,?2)Xi?N(?,?2). Suppose that x¯=11x¯=11, s2=16s2=16, and ?=0.05?=0.05.

Assuming it is known that ?2=4?2=4, test H0:?=12H0:?=12 versus Ha:?<12Ha:?<12.

Calculate the probability of type II error of in fact ?=10.5?=10.5

Hint: P(Type II error)=?=?(z1????H0??true?/n???)P(Type II error)=?=?(z1????H0??true?/n)

Using the information in part (b), what sample size is needed for the power of the test (1??)(1??) to be equal to 0.90?