A factory tests all its products. The proportion of defective items is 0.02. The probability that the test will catch a defective product is 0.95. The test will also reject nondefective products with probability 0.01. (a) Given that a product passes the test, what is the probability that it is defective? (b) Given that the product does not pass the test, what is the probability that the product is defective?

bag contains 7 red marbles, 5 white marbles, and 8 blue marbles. You draw 5 marbles out at random, without replacement. What is the probability that all the marbles are red?

The probability that all the marbles are red is  .

What is the probability that exactly two of the marbles are red?


What is the probability that none of the marbles are red?

Student scores on Professor Combs' Stats final exam are normally distributed with a mean of 77 and a standard deviation of 7.5 Find the probability of the following: **(use 4 decimal places)** a.) The probability that one student chosen at random scores above an 82. b.) The probability that 20 students chosen at random have a mean score above an 82. c.) The probability that one student chosen at random scores between a 72 and an 82. d.) The probability that 20 students chosen at random have a mean score between a 72 and an 82.

q2. World class marathon runners are known to run that distance (26.2 miles) in an average of 143 minutes with a standard deviation of 13 minutes.

If we sampled a group of world class runners from a particular race, find the probability of the following:

**(use 4 decimal places)**

a.) The probability that one runner chosen at random finishes the race in less than 137 minutes.

b.) The probability that 10 runners chosen at random have an average finish time of less than 137 minutes.

c.) The probability that 50 runners chosen at random have an average finish time of less than 137 minutes.

A certain stock market had a mean return of 2.6% in a recent year. Assume that the returns for stocks on the market were distributed normally, with a mean of 2.6 and a standard deviation of 10. Complete parts (a) through (g) below.

a. If you select an individual stock from this population, what is the probability that it would have a return less than 0 (that is, a loss)?

The probability is (Round to four decimal places)

b. If you select an individual stock from this population, what is the probability that it would have a return between -11 and -19?

The probability is (Round to four decimal places)

c. If you select an individual stock from this population, what is the probability that it would have a return greater than -7?

The probability is (Round to four decimal places)

d. If you select a random sample of four stocks from this population, what is the probability that the sample would have a mean return less than 0 (a loss)?

e. If you select a random sample of four stocks from this population, what is the probability that the sample would have a mean return between -11 and -19?

The probability is (Round to four decimal places)

a. Major League Baseball now records information about every pitch thrown in every game of every season. Statistician Jim Albert compiled data about every pitch thrown by 20 starting pitchers during the 2009 MLB season. The data set included the type of pitch thrown (curveball, changeup, slider, etc.) as well as the speed of the ball as it left the pitcher’s hand. A histogram of speeds for all 30,740 four-seam fastballs thrown by these pitchers during the 2009 season is shown below, from which we can see that the speeds of these fastballs follow a Normal model with mean μ = 92.12 mph and a standard deviation of σ = 2.43 mph.

b. Scores for a common standardized college aptitude test are normally distributed with a mean of 480 and a standard deviation of 105. Randomly selected men are given a Test Preparation Course before taking this test. Assume, for sake of argument, that the preparation course has no effect.

If 1 of the men is randomly selected, find the probability that his score is at least 532.5.
P(X > 532.5) =
Enter your answer as a number accurate to 4 decimal places.

If 9 of the men are randomly selected, find the probability that their mean score is at least 532.5.
P(M > 532.5) =
Enter your answer as a number accurate to 4 decimal places.

c. A population of values has a normal distribution with μ=63.8μ=63.8 and σ=88.3σ=88.3. You intend to draw a random sample of size n=98n=98.

Find the probability that a single randomly selected value is less than 83.4.
P(X < 83.4) =

Find the probability that a sample of size n=98n=98 is randomly selected with a mean less than 83.4.
P(M < 83.4) =

Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

d. Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find the probability of obtaining a reading greater than 3.186°C. Round answers to 4 decimal places.

P(Z>3.186)=P(Z>3.186)=

e.

Assume that the readings at freezing on a bundle of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find the probability of obtaining a reading between -0.398°C and 1.554°C.

P(−0.398<Z<1.554)=P(-0.398<Z<1.554)=

Compute the z-score of pitch with speed 84.9 mph. (Round your answer to two decimal places.)



Approximately what fraction of these four-seam fastballs would you expect to have speeds between 88 mph and 93 mph? (Express your answer as a decimal, not a percent, and round to three decimal places.)



Approximately what fraction of these four-seam fastballs would you expect to have speeds below 88 mph? (Express your answer as a decimal, not a percent, and round to three decimal places.)



A baseball fan wishes to identify the four-seam fastballs among the slowest 4% of all such pitches. Below what speed must a four-seam fastball be in order to be included in the slowest 4%? (Round your answer to the nearest 0.1 mph.)

mph

1. Thirty-two small communities in Connecticut (population near 10,000 each) gave an average of x = 138.5 reported cases of larceny per year. Assume that σ is known to be 40.9 cases per year.

(a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(d) Compare the margins of error for parts (a) through (c). As the confidence levels increase, do the margins of error increase?

As the confidence level increases, the margin of error decreases.

As the confidence level increases, the margin of error increases.    

As the confidence level increases, the margin of error remains the same.

(e) Compare the lengths of the confidence intervals for parts (a) through (c). As the confidence levels increase, do the confidence intervals increase in length?

As the confidence level increases, the confidence interval decreases in length.

As the confidence level increases, the confidence interval increases in length.    

As the confidence level increases, the confidence interval remains the same length.

2. Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 40 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 7.50 ml/kg for the distribution of blood plasma.

(a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.)

(b) What conditions are necessary for your calculations? (Select all that apply.)

n is large

the distribution of weights is normal

σ is known

σ is unknown

the distribution of weights is uniform

(c) Interpret your results in the context of this problem.

1% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.99.   

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.01.

99% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.

(d) Find the sample size necessary for a 99% confidence level with maximal margin of error E = 2.70 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.)
_____ male firefighters

3. Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 10 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.32 gram.

(a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

(b) What conditions are necessary for your calculations? (Select all that apply.)

n is large

uniform distribution of weights

σ is known

normal distribution of weights

σ is unknown

(c) Interpret your results in the context of this problem.

The probability to the true average weight of Allen's hummingbirds is equal to the sample mean.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.    

There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.

There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.14 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)
hummingbirds

1. The ReservedRoom Class:

Each ReservedRoom entry has three (data) fields:

• roomID (string) of the classroom e.g. W330, W350, W361, etc.
• courseID (string) which stores which course reserves this room
• time (int): which stores the start time of the reserved slot i.e. 1820 means the reservation starts at 18:20

Thus, the ReservedRoom class will consist of three data members (roomID and courseID, time), in addition to the associated methods needed as follows.

1. The constructor ReservedRoom (r, c, t): constructs a ReservedRoom object with the given parameters.
2. String getRoom(): returns the roomID field.
3. String getCourse(): returns the courseID field.
4. int getTime(): returns the time field.
5. String toString(): returns a String representation of this ReservedRoom entry.

Implement the ReservedRoom ADT using Java programming language.

The RoomsBoard Class:

RoomsBoard is a singly-linked list of ReservedRoom. The class RoomsBoard supports the following operations:

1. void add(ReservedRoom r): adds a new ReservedRoom object to RoomsBoard. The ReservedRoom object should be inserted into the RoomsBoard list such that the nodes of the list appear in non-decreasing order of time.
2. void remove(String roomID): removes all occurrences of the room with this ID.
3. void remove_all_but(String roomID): removes all the reserved rooms except the room with this ID.
4. void removeAll(): clears the RoomsBoard by removing all rooms entries.
5. void split( RoomsBoard board1, RoomsBoard board2): splits the list into two lists, board1 stores the list of reserved rooms before 12:00 and board2 stores the list of rooms after 12:00.
6. void listRooms(): prints a list of the reserved rooms in the roomsboard, in non-decreasing order.
7. Void listReservations( roomID): prints the list of reservations for this room in the roomsboard, in non-decreasing order.
8. int size(): returns the number of reserved rooms stored in the roomsboard.
9. boolean isEmpty(): returns true if the RoomsBoard object is empty and false otherwise.

Implement the RoomsBoard ADT with a singly-linked list using Java programming language. The implementation must utilize the generic class Node as described in Section 3.2.1 of your text. Figure 1 shows a sample RoomsBoard object.

1. The Menu-driven Program:

Write a menu-driven program to implement a roomsboard for a classroom reservations system. The menu includes the following options:

1. Add new room.
   The program will prompt the user to enter the roomID, courseID and time of the new reserved room, then will call the add() method from the RoomsBoard class.
2. Remove all occurrences of a room.
   The program will prompt the user to input the room ID to be removed, then call the remove() method from the RoomsBoard class.
3. Remove all rooms except a room.
   The program will prompt the user to input the room ID to be kept, then call the remove_all_but () method from the RoomsBoard class.
4. Clear the RoomsBoard.

The program will call the removeAll() method from the RoomsBoard class.

5. Split the list of rooms into two lists by calling the method split() from the RoomsBoard class.
6. Display RoomsBoard.
   The program will call the listRooms() method from the RoomsBoard class.
7. Display the reservations of a room.
   The program will call the listReservations() method from the RoomsBoard class to display all reservations of this room with the given roomID.
8. Exit.

The program should handle special cases and incorrect input, and terminate safely. Examples include (but not limited to) the following:

1. Removing a RoomsEntry object from an empty RoomsBoard object.
2. Removing a ReservedRoom object with a roomID that is not in the RoomsBoard object.
3. Displaying an empty RoomsBoard object.
4. Entering an incorrect menu option.

In a certain​ country, the true probability of a baby being a

girl

is

0.461

Among the next

four

randomly selected births in the​ country, what is the probability that at least one of them is a

boy​?

The probability is.

Consider a security with the stock prices

S(1) =

:

80 with probability 1/8

90 with probability 2/8

100 with probability 3/8

110 with probability 2/8

(a) What is the current price of the stock for which the expected return

would be 12%?

(b) What is the current price of the stock for which the standard deviation

would be 18%?