+1-415-315-9853

info@mywordsolution.com

## Statistics

1) The t and F Distributions

The T random variable with v degrees of freedom has the distribution

Show that the random variable F = T2 has the F distribution with 1 and v degrees of freedom.

2) An Unbiased Estimate of σ

The chi distribution models the square root of a chi-square random variable:

Y= √x      X~ x2 v

(a) Find the pdf of the chi distribution.

(b) We know that the sample variance from a normal distribution follows a chi-square distribution

Find the expected value of the sample standard deviation, s, and suggest an adjustment that would make it unbiased. You may find the following formulas useful:

3) A Catch-and-Release Estimate

A park has N raccoons of which 10 were previously captured and tagged. Suppose that 20 raccoons are captured. Find the probability that n = 5 of these are found to be tagged. Denote this probability by p(N).

(a) Find the value of N that maximizes p(N); this is called a maximum likelihood estimate. Hint: compare the ratio p(N)/p(N -1) to unity.

(b) Plot the maximum likelihood estimate of N against varying values of n, from 1 to 10.

4) Finite Population Correction Factor

For a finite population of size N with mean µ and variance σ2, it can be shown that the covariance of any two observations in a sample is

and that the sample variance is slightly biased

This means that the variance of the sample mean, when drawn from a finite population, includes a covariance term. This gives rise to the finite population correction factor. Use these relationships to show that the variance of the sample mean is

and that this is estimated by

5) Zero-Intercept Regression

Some phenomena follow a linear model which has an inherent zero response for a zero predictor, so the candidate regression model is

y=βx+ε

(a) Find the ordinary least squares estimator for β  by minimizing the sum of squared errors

(b) It seems reasonable that the variance around the regression line would increase as the predictor increases (think of the line swinging around the fixed "pivot" at the origin). If the error was normally distributed, this could give rise to the conditional distribution

Assuming this distribution, find the maximum likelihood estimator for β .

describe why this is called a ratio estimator.

(c) for the following data, estimate β using both your OLS and MLE estimators.

x  0.5 1.5 3.2 4.2 5.1 6.5

y  1.3 3.4 6.7 8.0 10.0 13.2

6) A Truncated Distribution

Students in several statistics classes were asked to complete a problemnaire. One of the quantities asked was the number of siblings a student had. This is a summary of the responses:

siblings    frequency

0                  4

1                 22

2                 22

3                 11

4                  8

5                  3

6                  3

12               1

20               1

6.1) Problem. Use the sibling data to estimate a distribution for the number of children in a family. Obviously, the number of children is one more than the number of siblings. However, there is a selection bias in this measurement; families with no children cannot be reported this way! Therefore the data follows a zero-truncated Poisson distribution (ZTPD)

(a) Find the MLE for the parameter λ of the ZTPD.

(b) What are the mean and variance of the ZTPD? Rather than directly calculating the moments, you might find it simpler to use the probability generating function

Gk(z) = E[zk]

and then take advantage of the properties of the pgf:

E[K] = G'k(z = 1)   V ar[K] = G"k(z = 1) + G'k(z = 1)- [G'k(z=1)]2

(c) Estimate the mean for the number of children and determine whether the data fits a ZTPD with that mean.

7) Estimating with Confidence The times to failure (in hours) for a sample of n = 30 backup generators are

7494.7       8801.7     9990.7      11277.7     10173.3      7746.8

9003.6       8242.9     4532.2      12541.8      6766.9       9898.9

8922.0       13429.8   17623.5    9135.6        6029.8       9038.7

20972.0     7605.1     5396.6      7528.2        10330.6     6475.4

12390.9     9857.0     7067.6      9704.2         5055.8      9942.4

(a) Find the mean time to failure and a 95% confidence interval (use a t distribution).

(b) Find a 95% confidence interval for the standard deviation (start with a chi-square interval for the variance).

Statistics and Probability, Statistics

• Category:- Statistics and Probability
• Reference No.:- M9290

Have any Question?

## Related Questions in Statistics and Probability

### 1 if the independent variable has absolutely no effect on

1. If the independent variable has absolutely no effect on the dependent variable, will the means of the experimental conditions be the same? Why or why not? 2. An ANOVA for a one-way design partitions the total variance ...

### The course text notes that tensions and differences in the

The course text notes that tensions and differences in the 1770s led to the American Revolution that eventually produced independence for the United States from Great Britain. When the War for American Independence broke ...

### 1 what is the difference between descriptive and

1. What is the difference between descriptive and inferential statistics? 2. Describe the most common descriptive statistics. 3. What are standard scores, and how are they calculated? 4. What is meant by an unbiased esti ...

### 1 whats so bad about making a type i error2 how does the

1. What's so bad about making a Type I error? 2. How does the Bonferroni adjustment control for Type I error when many statistical tests are conducted? 3. Why do researchers use ANOVA rather than t-tests to analyze data ...

### Hello kitty company claims that the proportion of its

Hello Kitty Company claims that the proportion of its employees investing in individual investment accounts is higher than national proportion of 50%. A survey of 100 employees in Hello Kitty Company indicated that 58 of ...

### A company manufactuirng compter chip finds that 8 of all

A company manufactuirng compter chip finds that 8% of all chips manufactured are defective. Management is concerned that high employee turnover is partially responsible for the high defect rate. In an effort to decrease ...

### All the customers who entered a store on a particular day

All the customers who entered a store on a particular day were given a surgery to fill out concerning their opinions of the service at the store? What type of sample is this? I can't decide between voluntary response or ...

### I love juliet and i know that juliet loves me but i do not

I love Juliet, and I know that Juliet loves me, but I do not know if Juliet knows that I love her. Provide an Aumann model of incomplete information that describes this situation, and specify a state of the world in that ...

### A sample of 19 school children was randomly selected and

A sample of 19 school children was randomly selected and their age, height and weight was recorded for each child. The results are given below. The goal is to predict the weight of the child by using the height and age. ...

### 1 what does the error of estimation tell us about the

1. What does the error of estimation tell us about the results of a study conducted using probability sampling? 2. What happens to the error of estimation as one's sample size increases?

• 13,132 Experts

## Looking for Assignment Help?

Start excelling in your Courses, Get help with Assignment

Write us your full requirement for evaluation and you will receive response within 20 minutes turnaround time.

### Section onea in an atwood machine suppose two objects of

SECTION ONE (a) In an Atwood Machine, suppose two objects of unequal mass are hung vertically over a frictionless

### Part 1you work in hr for a company that operates a factory

Part 1: You work in HR for a company that operates a factory manufacturing fiberglass. There are several hundred empl

### Details on advanced accounting paperthis paper is intended

DETAILS ON ADVANCED ACCOUNTING PAPER This paper is intended for students to apply the theoretical knowledge around ac

### Create a provider database and related reports and queries

Create a provider database and related reports and queries to capture contact information for potential PC component pro

### Describe what you learned about the impact of economic

Describe what you learned about the impact of economic, social, and demographic trends affecting the US labor environmen