Probstat/homework 2

This is part of probstat.

Deadline: December 7th, 2014.

Part 1: Review of the normal distributions.

• work on Week 8 practice 2, basic practice, questions 2,4,6,8, and 10.

Part 2: Distribution of sample means.

2.1 Consider a discrete random variable $Y$ where $P\{Y=0\}=0.3$, $P\{Y=1\}=0.3$, and $P\{Y=2\}=0.4$.

(a) If we take a sample of size 2, and let ${\bar {Y}}$ be the sample mean. Describe the distribution of ${\bar {Y}}$.

(b) If we take a sample of size 3, and let ${\bar {Y}}$ be the sample mean. Describe the distribution of ${\bar {Y}}$. (Note: you can write a computer program to help you calculate the distribution. But if you don't, thinking about 9 cases is not too bad.)

Hint: To describe the distribution of normally distributed random variables, you can just specify their means and their variances (or s.d.'s).

2.2 Suppose that the population is normally distributed with mean $\mu =10$ and variance $\sigma ^{2}=4$. Let ${\bar {X}}$ be the mean of a sample of size 10. Describe the distribution of ${\bar {X}}$.

2.3 Find $P\{{\bar {X}}>11\}$. (Hint: use the standard normal table to find the answer.)

2.4 Find $P\{|{\bar {X}}-10|>1\}$. (Hint: use the standard normal table to find the answer.)

Part 3: Estimations and confidence intervals.

1. You take a sample of size 15 from a population with the known variance of 6. Let $\mu$ be the mean of the population. What is the probability that the sample mean ${\bar {X}}$ is such that $P\{|{\bar {X}}-\mu |>1\}$.

Note: this question has be corrected.

2. You take a sample of size 15 from a population with the known variance of 6. Let $\mu$ be the mean of the population. What is the probability that the sample mean ${\bar {X}}$ is such that $P\{{\bar {X}}<\mu -1.5\}$.

Note: this question has be corrected.

3. You take a sample of size 20 from a population with the known variance of 5. Let ${\bar {X}}$ be the sample mean. Find the confidence interval for the confidence level of 90% and 95%.

4. You take a sample of size 10 from a population with an unknown variance. Let ${\bar {X}}$ be the sample mean and let $S^{2}$ be the sample variance. Find the confidence interval for the confidence level of 95% and 99%. (Hint: use t-distribution.)

5. The following sample is taken from a population whose standard deviation is 10. Find the sample mean together with the confidence interval with the confidence level of 90%.

254.8364581731
232.9768161925
244.2857506108
241.4092424584
238.7415013585
238.8916551149
225.6090880531
243.1193936041
227.8234629926
235.6030158527

6. The following sample is taken from a population whose standard deviation is unknown. Find the sample mean together with the confidence interval with the confidence level of 90%. Is it possible to say that the population mean is greater than 0 with confidence level 90%? Why?

29.6788710493
30.2171658494
10.3209296273
-30.8989271632
-9.6774494881
19.862254862
1.6794611651
-4.5608921106
3.8877580944
-3.1861137644
51.6477880725
17.9539116735
21.3801555032
32.6735671462
6.1932339461