1. A really bad carton of 18 eggs contains 7 spoiled eggs. An unsuspecting chef picks 4 eggs at random for this "Mega-Omelet Surprise". Let x be the number of unspoiled eggs in a sample of 4 eggs. Find the probability that the number of unspoiled eggs among the 4 selected is exactly 4.Round your answer to the nearest 4 decimal places.

2. Let x be a continuous random variable that has a normal distribution with a mean of 25 and a standard deviation of 5. Find the probability, rounded to the nearest 4 decimal places, that x is less than or equal to 20.

An automobile manufacturer has determined that 30% of all gas tanks that were installed on its 2007 compact model are defective.

a. If 15 of the cars are recalled, what is the expected number (mean) that will need new gas tanks?

b. If 15 of the cars are recalled, what is the standard deviation of the number of defective gas tanks?

Use Binomial Tables in your text to answer the following:

c. If 15 of the cars are recalled, what is the probability that 10 of the 15 will need new gas tanks?

d. If 15 of the cars are recalled what is the probability that fewer than 3 of the 15 will need new gas tanks?

The average number of customers visiting the science center was 800 per day last year and the populations standard deviation is 250 customers per day.

1. In a span of a month, i.e. 30 days, write out the distribution of the sample mean

2. What is the probability that the sample mean is over 275 customers per day in a month?

3. What is the probability that the sample mean is less than 275 customers per day in a month?

4. A good month means the average number of customers is more than the average of 95% of the other month. Determine the criteria of a good month.

Sampling distribution question

1. State the definition of sampling distribution?
2. Let’s assume that we have a Bernoulli random variable X,

X = a with probability 0.78,

X = b with probability 0.22,  

Where a and b are the third and fourth digit of your student number, respectively.(a=9,b=9) Develop a sampling distribution for sample means of the Bernoulli distribution when the sample size is 6

3. What is the expected value of sample means in question 3.b? What is the variance of the sample means?
4. If the sample size increases to a large number n (n is greater than 30), how are the sample means distributed?

1a. Suppose I have a gaming web site that can only handle 10 players at the same time, or else my server will crash. I have 50 users. Each user is online and playing the game 20% of the time. What is the probability that my server will crash. 1b Now suppose that it is acceptable if the crashing probability is less than 1%. What is the maximum number of users my server can handle? 1c. What is the maximum number of users I can support if my server is twice as large and can handle 20 simultaneous players?

One of the quirks of the ground is occasional acid fog. Only 150 grounders are immune to acid fog out of a total of 3000 grounders. For a random group of 250 grounders in an acid fog;

A. (6 points) What is the expected value and standard deviation of the number of survivors? (HINT: It might be useful to think about the Binomial distribution here.) (Your answer here)

B. (8 points) what is the approximate probability that the number of survivors is at least 20? (Your answer here)

C. (6 points) Verify the conditions needed to compute the probability in Part (B) above. (Your answer here)

The Genetics & IVF Institute in Fairfax, Virginia, have developed a technique called MicroSort that claims to increase the chances of a couple to have a baby of a specified gender. The method for increasing the likelihood of a girl is referred to as XSORT and for increasing the likelihood of a boy is called YSORT. In a clinical trial, 945 couples requested baby girls and therefore were given the sperm form the XSORT method. Out of the 945 couples, 879 of them had a baby girl. In this case study, we will work toward answering the question: “Is the XSORT method effective in increasing the chance of a baby girl being born?” Questions with lines are expecting only an answer on the line while those without lines expect a sentence or two of explanation or computations.

1) What would be the expected value for the number of girls born out of 945 births in the XSORT method had no effect?

2) Using your answer to number (1), what is your initial option on whether the XSORT method is effective? (No statistics/calculations needed. Simply state your opinion and why.)

Before we continue to create the statistical argument to answer our question of the effectiveness of the XSORT method, we will take some time working with only two births. 3) So, the discrete random variable for our probability model would be X = number of ___________ in two births. (Fill in the blank.)

4) State the probability distribution for the number of (answer to (3)) in two births. Be sure to show that the two properties required for a probability distribution are satisfied.

5) Create the associated probability histogram.

**** LOOKING TO SOLVE THE SECOND PART TO THIS 2 PART QUESTION; PROBLEM 4 ONLY PLEASE ****

Automobiles arrive at the drive-through window at a post office at the rate of 4 every 10 minutes. The average service time is 2 minutes. The Poisson distribution is appropriate for the arrival rate and service times are exponentially distributed.
a) What is the average time a car is in the system? 10 minutes
b) What is the average number of cars in the system? 4
c) What is the average time cars spend waiting to receive service? 8 minutes
d) What is the average number of cars in line behind the customer receiving service? 3.2
e) What is the probability that there are no cars at the window? 0.2
f) What percentage of the time is the postal clerk busy? 80%
g) What is the probability that there are exactly two cars in the system? 0.128

Problem-4:
For the post office in the previous problem, a second drive-through window is being considered. A single line would be formed and as a car reached the front of the line it would go to the next available clerk. The clerk at the new window works at the same rate as the current one.
• What is the average time a car is in the system?
• What is the average number of cars in the system?
• What is the average time cars spend waiting to receive service?
• What is the average number of cars in line behind the customer receiving service?
• What is the probability that there are no cars in the system?
• What percentage of the time are the clerks busy?
• What is the probability that there are exactly two cars in the system?

The number of times a machine broke down each week was observed over a period of 100 weeks and recorded as shown in the table below. Complete parts a and b below.
Number of
Breakdowns
0
1
2
3
4
5 or More

Number of
weeks
9
22
32
26
6
5

a. Using alphaequals?0.10, perform a? chi-square test to determine if the population distribution of breakdowns follows the Poisson probability distribution.
What is the null? hypothesis, Upper H 0??
A.
The distribution of breakdowns differs from the expected distribution.
B.
The distribution of breakdowns follows the Poisson probability distribution.
C.
The distribution of breakdowns follows the normal probability distribution.
D.
The distribution of breakdowns follows a uniform probability distribution.

What is the alternate? hypothesis, Upper H 1??
A.
The distribution of breakdowns follows a uniform probability distribution.
B.
The distribution of breakdowns follows the Poisson probability distribution.
C.
The distribution of breakdowns differs from the claimed or expected distribution.
D.
The distribution of breakdowns follows the normal probability distribution.

Determine the estimated mean of the frequency distribution. The average number of breakdowns per week over this period is
?(Type an integer or a? decimal.)

Calculate the test statistic.
(Round to two decimal places as? needed.)

Determine the critical? value, (Round to three decimal places as? needed.)

Choose the correct rejection region below.
A.
chi squared less than or equals chi Subscript alpha Superscript 2
B.
chi squared greater than or equals chi Subscript alpha Superscript 2
C.
chi squared less than chi Subscript alpha Superscript 2
D.
chi squared greater than chi Subscript alpha Superscript 2

Draw a conclusion.
?
the null hypothesis. Based on the? results, it is reasonable to assume the distribution of breakdowns
?
does not follow
follows
the Poisson probability distribution for making business decisions.
b. Determine the? p-value and interpret its meaning.
?p-valueequals
nothing ?(Round to three decimal places as? needed.)
Interpret the? p-value.
The? p-value is the probability of observing a test statistic
?
less than
equal to
greater than
the test? statistic, assuming
?
the distribution of the variable is the same as the given distribution.
at least one expected frequency differs from 5.
the distribution of the variable differs from the given distribution.
the expected frequencies are all equal to 5.
the distribution of the variable is the normal distribution.
the distribution of the variable differs from the normal distribution.
At alphaequals?0.10, what is the correct? conclusion?
?
Reject
Do not reject
the null hypothesis. There is
?
insufficient
sufficient
evidence to conclude that the population distribution of breakdowns is not Poisson.

A retail store has implemented procedures aimed at reducing the number of bad checks cashed by its cashiers. The store's goal is to cash no more than eight bad checks per week. The average number of bad checks cashed is 12 per week. Let x denote the number of bad checks cashed per week. Assuming that x has a Poisson distribution:

(a) Find the probability that the store's cashiers will not cash any bad checks in a particular week. (Round your answer to 4 decimal places. Leave no cells blank - be certain to enter "0" wherever required.)

(b) Find the probability that the store will meet its goal during a particular week. (Round your answer to 4 decimal places. Leave no cells blank - be certain to enter "0" wherever required.)

(c) Find the probability that the store will not meet its goal during a particular week. (Round your answer to 4 decimal places. Leave no cells blank - be certain to enter "0" wherever required.)

(d) Find the probability that the store's cashiers will cash no more than 10 bad checks per two-week period. (Round your answer to 4 decimal places. Leave no cells blank - be certain to enter "0" wherever required.)

(e) Find the probability that the store's cashiers will cash no more than five bad checks per three-week period. (Round your answer to 4 decimal places. Leave no cells blank - be certain to enter "0" wherever required.)