A traffic light at a certain intersection is green 50% of the time,yellow 10% of the time, and red 40% of the time. A car approaches this intersection once each day. We would like to know about the number of days that pass up to and including the first time the car encounters a red light. Assume that each day represents an independent trial.

a) Define the random variable of interest, it support, and parameter values.

b) What is the probability that exactly three days passed up to and including the first time the car encounters a red light?

c) Find P(X<=3).

d) Find the average number of days that passes up to and including the first time the car encounters a red light.

After 100 throws of a 6-sided dice the following results were obtained

Is this data consistent with a fair dice (the one where each side comes up with equal probability) at a = 0.1. (13 points)

a. State the hypotheses and identify the claim. (2 points)

b. Calculate the predicted values. (2 points)

b. Compute the test value. (3 points)

c. Find the critical value. (3 points)

d. Make the decision to reject or not reject the null hypothesis. Justify. (1 point)

e. Summarize the results. (2 points)

Consider a statistical expirement of flipping a pair of fair coins simultaneously. Let X & Y be the number heads in flipping each coin.

Define a joint density function Z = XY. (I provided the answers in bold but I need help understanding how it is solved. Thank you in advance).

a) The number of possible distinct values of Z is: 2

b) The probability of Z = 0 is: 3/4

c) The mean of Z is: 1/4

d) The standard deviation of Z is: sqrt(3/16)

e) The correlation coefficient of X and Y is: 0

f) Which RV follows a uniform distribution among all three random variables, X, Y, and Z? X & Y

Start StatCrunch and make the following sequence selection: Applets -> Distribution demos. Next select "Binomial" and click "Compute!". In the resulting popup window experiment by using the sliders to assign approximately 0.5 to p and successively assign the values 20, 30 and 40 to n. Discuss what you see in the subsequently drawn Binomial Distribution defined by your specified values for n and p. What value on the x axis (horizontal axis) does the top of the hump of the curve correspond to. Next set p and n to their extreme values? Discuss what you observed using the fact that the x-axis represents the number of successes and the height of the vertical lines represent the probability of getting x number of successes.

A company would like to implement its inventory of smartphones as a doubly linked list, called

MobileList.

1. Write a Mobile node node class, called MobileNode, to hold the following information about a

smartphone:

• code (as a String)

• brand (as a String)

• model (as a String)

• price (as int)

MobileNode should have constructors and methods (getters, setters, and toString()) to manage

the above information as well as the link to next and previous nodes in the list.

2. Write the MobileList class to hold objects of the class MobileNode. This class should define:

• Two instance variables first and last to keep the reference (address) of the first and last

nodes of the list.

• The MobileList class should implement the following interface:

public interface MList {

public boolean isEmpty();

// returns true if the list is empty, false otherwise

public int size();

// returns the number of items in the list

public MobileNode getNodeAt(int index);

//returns the MobileNode object at the specified index

public void addFirst(MobileNode item);

// adds a Mobile node at the front of the list

public void addLast(MobileNode item);

// adds a Mobile node at the end of the list

public void addAt(int index, MobileNode item);

// adds a Mobile node to the list at the given index

public String removeAt(int index);

// removes the Mobile node from the list that has the given

// index

public String remove(MobileNode item);

// removes the first item in the list whose data equals

// the given item data

public MobileNode[] searchPriceGreaterThan(int p);

//search and return an array of the set of MobileNode items

//having a price greater than p

public double averagePrice();

// return the average price of the mobile nodes in the list

public double averageBrandPrice(String brand);

// return the average price of the mobile nodes in the list

// from the brand given as a parameter (e.g., “Samsung” or

// “samsung”)

@Override

public String toString();

// implement a toString() method that prints the list in the

// format:

//[ size: the_size_of_the_list

// Mobile node1,

// Mobile node2,

//.... ]

}

3. Write a TestMobileList class to test the class MobileList. This class should have a main method

in which you perform the following actions:

• Create a MobileList object,

• Insert 10 MobileNode objects into the created list (from some brands like “Apple”,

“Samsung”, “Huwaei”, “Sony”),

• Print the content of your list,

• Find out in the list the items that have a price greater than 3000. Print them out.

• Remove the first element of the list

• Remove the item at index 3

• Print again the content of your ist,

• Print out the average price of all mobiles in the list

• Print out the average price of all mobiles in the list from “Apple”.

For each operation above, print a small message that describes the operation you are doing.

For example:

System.out.println(“Insertion of 10 Mobile nodes in the list”);

1. Consider the data set below.determine if this data comes from a normally distributed population.

1. Draw a histogram and determine if it seems to be mound shaped
2. Determine the number of outliers for your dataset
3. Draw a normal probability plot.

You should comment on whether theses measure suggest normality or not. Use all the measures in your discussion

This data is the number of deaths each year by tornados in the past 30 years

2. Consider the data set below. Your job is to determine if this data comes from a normally distributed population. Remember this means that you should do the following

4. Draw a histogram and determine if it seems to be mound shaped
5. Determine the number of outliers for your dataset
6. Draw a normal probability plot.

You should comment on whether theses measure suggest normality or not. Use all the measures in your discussion

This data is body temperatures from randomly selected individuals at the same time of day (measures in degrees Fahrenheit)

Use the P-value Approach for all hypothesis tests. Assume that all samples are randomly obtained.

The first significant digit in any number is: 1, 2, 3, 4, 5, 6, 7, 8, and 9. Though we may think that each digit would appear with equal frequency, this is not true. Physicist, Frank Benford, discovered that in many situations where counts accumulate over time that the first digit in the count follows a particular pattern. The probabilities of occurrence to the first digit in a number, known as Benford’s Law, are shown below.

Bloomberg’s web site (www.bloomberg.com) gives information on the stock price and trading volume of the members of the S&P 500 stock index. Suppose a random sample of 100 stocks on a given day were selected from the site, and the first digit of the daily stock volume (total number of shares traded in a given day) was recorded below.

Using a 10% level of significance, test whether the first digits in the stock price follow the distribution of probabilities given by Benford’s Law. Be sure to verify the requirements for the test.

A newsboy sells newspapers and his goal is to maximize profit. He kept a record of his sales for 125 days with the following result. His ordering policy is to order an amount each day that is equal to the previous day's demand. A newspaper costs the carrier 50 cents and he sells it for $1.00. Unsold papers are returned and he receives 25 cents (for a loss of 25 cents).

Use the information and random numbers given in the table below to simulate the sale of newspapers for 10 days.

12. After completing the simulation, determine his total revenue for the ten days. _____

13. After completing the simulation, determine the monetary losses that result from unmet demand and unsold papers.   _____

Mary has her own business, a nail salon that she runs out of her home.She is the only nail technician but serves many clients in one day. The following tables provide information about time between arrivals and service time required for a manicure. Assume the only service offered is a manicure.

The first random number generted for arrivals is used to tell us then the first customer arrives after the shop opens. the second random number generated for service is used to tell us how long the service took.

Answer the following questions using the charts above:

1) What time does the first customer arrive if the shop opens at 8 am?

2) What number customer will be the first to wait?

3) How long does the second customers appointment take?

________a number that is used to represent a population characteristic and that generally cannot be determined easily

________a method for selecting a sample and dividing the population into groups; use simple random sampling to select a set of groups. Every individual in the chosen groups is included in the sample.

________a method for selecting a sample used to ensure that subgroups of the population are represented adequately; divide the population into groups. Use simple random sampling to identify the number of individuals from each group.

_____ the set of all possible outcomes of an experiment

________a numerical characteristic of the sample

________all individuals, objects, or measurements whose properties are being studied

• _______ deals with estimating a population parameter based on a sample statistic.
• _______ the decision is to reject the null hypothesis when, in fact, the null hypothesis is true.
• _______ probability of a Type I error.
• _______ is a statement about the value of a population parameter
• _________ states that if the size n of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population.
• _______ the probability that an event will happen purely by chance assuming the null hypothesis is true.
• _______ the decision is not to reject the null hypothesis when, in fact, the null hypothesis is false.
• _______ is a single number computed from a sample and used to estimate a population parameter.
• ____________ is an interval estimate for an unknown population parameter.

Cluster Sampling

Sample Space

Population

Stratified Sampling

Parameter

Statistic

1. p-value
2. Confidence Interval
3. Inferential Statistics
4. Level of Significance
5. Type II Error
6. Hypothesis
7. Central Limit Theroem
8. Type I Error
9. point estimate