ผลต่างระหว่างรุ่นของ "Probstat/homework 2"

รุ่นแก้ไขปัจจุบันเมื่อ 06:01, 3 ธันวาคม 2557

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

@@ แถว 1: / แถว 1: @@
-: ''This is part of [probstat]]''.
+: ''This is part of [[probstat]]''.
 '''Deadline:''' December 7th, 2014.
@@ แถว 9: / แถว 9: @@
 == Part 2: Distribution of sample means. ==
-.1 Suppose that the population is normally distributed with mean <math>\mu=10</math> and variance <math>\sigma^2 = 4</math>.  Let <math>\bar{X}</math> be the mean of  a sample of size 10. Describe the distribution of <math>\bar{X}</math>.
+.1 Consider a discrete random variable <math>Y</math> where <math>P\{Y=0\} = 0.3</math>, <math>P\{Y=1\} = 0.3</math>, and <math>P\{Y=2\} = 0.4</math>.
-.2 Find <math>P\{\bar{X} > 11\}</math>.  (Hint: use the standard normal table to find the answer.)
+(a) If we take a sample of size 2, and let <math>\bar{Y}</math> be the sample mean.  Describe the distribution of <math>\bar{Y}</math>.
-.3 Find <math>P\{|\bar{X} - 10 | > 1\}</math>.  (Hint: use the standard normal table to find the answer.)
+(b) If we take a sample of size 3, and let <math>\bar{Y}</math> be the sample mean.  Describe the distribution of <math>\bar{Y}</math>.  (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 Suppose that the population is normally distributed with mean <math>\mu=10</math> and variance <math>\sigma^2 = 4</math>.  Let <math>\bar{X}</math> be the mean of  a sample of size 10. Describe the distribution of <math>\bar{X}</math>.
+.3 Find <math>P\{\bar{X} > 11\}</math>.  (Hint: use the standard normal table to find the answer.)
+.4 Find <math>P\{|\bar{X} - 10 | > 1\}</math>.  (Hint: use the standard normal table to find the answer.)
 == Part 3: Estimations and confidence intervals. ==
+. You take a sample of size 15 from a population with the known variance of 6.  Let <math>\mu</math> be the mean of the population.  What is the probability that the sample mean <math>\bar{X}</math> is such that <math>P\{|\bar{X} - \mu| > 1\}</math>.
+'''Note:''' this question has be corrected.
+. You take a sample of size 15 from a population with the known variance of 6.  Let <math>\mu</math> be the mean of the population.  What is the probability that the sample mean <math>\bar{X}</math> is such that <math>P\{\bar{X} < \mu - 1.5\}</math>.
+'''Note:''' this question has be corrected.
+. You take a sample of size 20 from a population with the known variance of 5.   Let <math>\bar{X}</math> be the sample mean.  Find the confidence interval for the confidence level of 90% and 95%.
+. You take a sample of size 10 from a population with an unknown variance.   Let <math>\bar{X}</math> be the sample mean and let <math>S^2</math> be the sample variance. Find the confidence interval for the confidence level of 95% and 99%.  (Hint: use t-distribution.)
+. 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%.
+<pre>
+.8364581731
+.9768161925
+.2857506108
+.4092424584
+.7415013585
+.8916551149
+.6090880531
+.1193936041
+.8234629926
+.6030158527
+</pre>
+. 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?
+<pre>
+.6788710493
+.2171658494
+.3209296273
+-30.8989271632
+-9.6774494881
+.862254862
+.6794611651
+-4.5608921106
+.8877580944
+-3.1861137644
+.6477880725
+.9539116735
+.3801555032
+.6735671462
+.1932339461
+</pre>

ผลต่างระหว่างรุ่นของ "Probstat/homework 2"

รุ่นแก้ไขปัจจุบันเมื่อ 06:01, 3 ธันวาคม 2557

Part 1: Review of the normal distributions.

Part 2: Distribution of sample means.

Part 3: Estimations and confidence intervals.

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