Prove that for a Markov chain on a finite state space, no states are null recurrent.

Your poll of 620 randomly selected residents of a Chicago suburb indicates that 32% of them would support the introduction of a halfway house for drug addicts in their community. Match the following elements needed to construct a 95% confidence interval.

1)Lower limit--------------->A).037

2)Margin of error----------> B)1.96

3)Critical value-----------> C).019

4)Standard error for proportions----------> D).283

1 point) If xx is a binomial random variable, compute P(x)P(x) for each of the following cases:

(a)  P(x≤1),n=5,p=0.3P(x≤1),n=5,p=0.3

P(x)=P(x)=

(b)  P(x>3),n=4,p=0.1P(x>3),n=4,p=0.1

P(x)=P(x)=

(c)  P(x<3),n=7,p=0.7P(x<3),n=7,p=0.7

P(x)=P(x)=

James was eating a bag of candies that came in eight different colors. He noticed that there appeared to be far fewer green candies than any of the other colors and wondered if the true proportion of green candies was lower than the 12.5% that would be expected if all of the candies came in even amounts. For the sake of statistics, he decided that he would need to buy more candy to test his hypothesis. James randomly selected several bags and candies and recorded the color of each piece of candy. He found that out of the first 400 candies that he chose, 39 of them were green.

James conducts a one-proportion hypothesis test at the 5% significance level, to test whether the true proportion of green candies was lower than 12.5%.

(a) H0:p=0.125; Ha:p<0.125, which is a left-tailed test.

(b) Use a TI-83, TI-83 plus, or TI-84 calculator to test whether the true proportion of green candies is less than 12.5%. Identify the test statistic, z, and p-value from the calculator output, rounding to three decimal places.

. Which   of   the   following   is   NOT   CORRECT   about   a   randomized   complete   block   experiment?  

(a)   Every   block   is   randomized   separately   from   every   other   block.  

(b)   Every   treatment   must   appear   at   least   once   in   every   block.  

(c)   Blocking   is   used   to   remove   the   effects   of   another   factor   (not   of   interest)   from   the   comparison   of  

levels   of   the   primary   factor.  

(d)   The   ANOVA   table   will   have   another   line   in   it   for   the   contribution   to   the   variability   from   blocks.  

(e)   Blocks   should   contain   experimental   units   that   are   as   different   as   possible   from   each   other.  

The table below shows performance data for 100 flights between cities A - G for some airline, including: date, flight #, origin, destination, # passengers flown (load), and tardiness (late, in hours).

write a VBA code (N=10000) to simulate the following:

1. (I7): number of flights which were late for at least 0.75 hrs.

2. (I10) average load of flights originated from C with load exceeding 250.

3. (I13): smallest tardiness of flights from B to E between 9/1/18 and 9/3/18.

4. (I16): total load of flights flown from A to C, D, and E on 9/4/18.

5. (I19): the flight # of the flight with the maximal load among all flights from A to G with tardiness less than 0.6 hrs.

The accuracy of a census report on a city in southern California was questioned by some government officials. A random sample of 1215 people living in the city was used to check the report, and the results are shown below.

Using a 1% level of significance, test the claim that the census distribution and the sample distribution agree.

(a) What is the level of significance?

State the null and alternate hypotheses.

H₀: The distributions are different.

H₁: The distributions are different.

H₀: The distributions are the same.

H₁: The distributions are different.    

H₀: The distributions are the same.

H₁: The distributions are the same.

H₀: The distributions are different.

H₁: The distributions are the same.

(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)

Are all the expected frequencies greater than 5?

Yes

No    

What sampling distribution will you use?

uniform

chi-square    

Student's t

normal

binomial

What are the degrees of freedom?

(c) Estimate the P-value of the sample test statistic.

P-value > 0.100

0.050 < P-value < 0.100    

0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories?

Since the P-value > α, we fail to reject the null hypothesis.

Since the P-value > α, we reject the null hypothesis.    

Since the P-value ≤ α, we reject the null hypothesis.

Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 1% level of significance, the evidence is sufficient to conclude that census distribution and the ethnic origin distribution of city residents are different.

At the 1% level of significance, the evidence is insufficient to conclude that census distribution and the ethnic origin distribution of city residents are different.  

Define the three types of hypothesis tests and explain why they are used. Give an example for each.

Assignment 1:

It is known that achievement test scores of all high school seniors in a state (in the US) have mean 60 and variance 64. On a large high school, located in a low socio economic area, a small group of senior students set out to investigate the performance of their own school. They collected a random sample of ??=100 achievement test scores from fellow students. Analyzing the results, it was found that the sample mean achievement test score was 58.

A) Is there evidence to suggest that this high school is performing poorly? Look at the hint below and provide an explanation.

(Hint: find the probability that the sample mean achievement test score is at most 58, assuming that the true mean score is 60. In the computation, use ? = 8)

A researcher investigates the factors that are associated with the salaries of professors who teach courses at a major university. The researcher gathers data about the subject area and the salary per course for a random sample of professors. Data are found in the file Academic Salaries by Subject Area.

a) State the null and alternate hypothesis we would run to determine if the average salaries of the professors is the same across all subjcet areas.

b) Run the appropriate test in Excel and show output. What conclusions can you make?

c) Obtain boxplots for these data, each individual data NPP, and a NPP for all residuals.

d) Are the assumptions of an ANOVA reasonably satisfied? Explain & Discussion in reference to the plots.

e) If there is a difference in salaries run a Tukey’s test to show how the salaries for the different subject areas compare to each other. Describe what the results of the Tukey’s test tell you.

Imagine a professor wants to examine if there is a relationship between gender and performance on a writing test. Thirty girls and thirty boys participated in his experiment. They were given a standard writing test and their grades were given as “outstanding”, “good”, “passing”, and “failing”. If the professor decided to use a Chi-square test to examine the relationship, how many degrees of freedom are there in this Chi-square test?

When using the Chi-square test, the probability of Type I error is____ its significance level.
a. Not related to   
b. Bigger than
c. Equal to

b. Smaller than

If the probability value of a Chi square of 3.2 with a degree of freedom of 3 is 0.368, then we should___.
a. Fail to reject the null hypothesis       
b. We cannot decide unless we get the information on t-value
c. Reject the null hypothesis       
d. We cannot decide unless we get the information on factors and levels

Describe the advantages and disadvantages of including interaction effects and polynomial terms in a multiple regression model.

Quick Start Company makes 12-volt car batteries. From historical data, the company knows that the life of such a battery is a normally distributed random variable with a mean life of 44 months and a standard deviation of 1010 months.
(a) What percentage of Quick Start 12-volt batteries will last between 32 months and 53 months?

(b) If Quick Start does not want to make refunds for more than 10% of its batteries under a full-guarantee policy, how long should the company guarantee the 12-volt batteries?

months
(c) Seventy-five 12-volt batteries are randomly selected n=75. What is the probability that the mean lifetime of the batteries in this sample will be between 42 and 43 months?

(Input answer to four decimal places)

Find the cumulative distribution function of X and draw its graph?

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability 0.3, and his second appointment will lead independently to a sale with probability 0.6. Any sale made is equally likely to be either for the deluxe model, which costs $1000, or the standard model, which costs $500. X is the total dollar value of all sales. Hint: you could find the probability mass function of X and use that.