Use Binomial Distribution (i believe)

1) Chromosome defect A occurs in only one out of 200 adult males. A random sample of 100 adult males is selected. Let the random variable ? represent the number of males in the sample who have this chromosome defect.

a) What are the mean and standard deviation of the random variable ??

b) What is the estimated probability that we observe 3 or more adult males with this chromosome defect? (calculate with and without continuity correction).

c) What is the exact distribution of ?? Use this to calculate the probability in part (b).

2) Chromosome defect A occurs in only one out of 200 adult males. A random sample of 100 adult males is selected. Let the random variable ? represent the proportion of males in the sample who have this chromosome defect.

a) What are the theoretical mean and standard deviation of ??

b) What is the distribution of sample proportion?

c) Compute the estimated probability that the sample proportion is 3% or more. Compare your results with the results of the previous question.

Use Binomial Distribution (i believe)

1) Chromosome defect A occurs in only one out of 200 adult males. A random sample of 100 adult males is selected. Let the random variable ? represent the number of males in the sample who have this chromosome defect.

a) What are the mean and standard deviation of the random variable ??

b) What is the estimated probability that we observe 3 or more adult males with this chromosome defect? (calculate with and without continuity correction).

c) What is the exact distribution of ?? Use this to calculate the probability in part (b).

2) Chromosome defect A occurs in only one out of 200 adult males. A random sample of 100 adult males is selected. Let the random variable ? represent the proportion of males in the sample who have this chromosome defect.

a) What are the theoretical mean and standard deviation of ??

b) What is the distribution of sample proportion?

c) Compute the estimated probability that the sample proportion is 3% or more. Compare your results with the results of the previous question.

Consider tossing a coin and rolling a four-sided die (with the numbers 1 through 4 printed on the sides).

(a) Describe the sample space.

(b) Whatistheprobabilityofrollingaheadsandanevennumber?

(c) What are the odds of rolling a heads and an even number?

(d) What is the probability of rolling a heads?

(e) What are the odds against rolling a heads?

Determine whether or not the following statements are correct or incorrect and explain why (Be thorough and clear in your explanation!):

(a) A person says “the odds of rolling a 1 on a standard six-sided die is 1/6.”

(b) In the past five seasons,cross town football riva ls the Quaker sand the Comet shave played

each other with the Comets winning twice and the Quakers winning three times. Someone

says “the odds in favor of the Quakers winning are 3:5.”

(c) If the odds in favor of an event occurring are A to B, then the probability of the event

occurring is A/(A+B).

(d) An event with probability 75% means that the event is three times more likely to occur than

not occur.

Commemorative coins are being struck at the local foundry. A gold blank (a solid
gold disc with no markings on it) is inserted into a hydraulic press and the obverse design is pressed
onto one side of the disc (this step fails with probability 0.15). The work is examined and if the
obverse pressing is good, the coin is put into a second hydraulic press and the reverse design is
imprinted (this step fails with probability 0.08). The completed coin is now examined and if of
sufficient quality is passed on for finishing (cleaning, buffing, and so on). Twenty gold blanks are
going to undergo pressing for this commemorative coin. Assume that all pressings are independent
of each other. What are the mean and variance of the number of good coins manufactured? If the blanks cost $300 each and the labor to produce the finished coins costs $3,000,
what is the probability that the production cost to make the 20 coins (labor and materials) can be
recovered by selling the coins for $500 each? (The $3,000 labor figure is the fixed cost to process
the 20 gold blanks -- some will be good some not.)

The manager of a computer retails store is concerned that his suppliers have been giving him laptop computers with lower than average quality. His research shows that replacement times for the model laptop of concern are normally distributed with a mean of 4.5 years and a standard deviation of 0.6 years. He then randomly selects records on 50 laptops sold in the past and finds that the mean replacement time is 4.3 years. Assuming that the laptop replacement times have a mean of 4.5 years and a standard deviation of 0.6 years, find the probability that 50 randomly selected laptops will have a mean replacement time of 4.3 years or less.

P(M < 4.3 years) =

Enter your answer as a number accurate to 4 decimal places.

Based on the result above, does it appear that the computer store has been given laptops of lower than average quality?

Yes. The probability of this data is unlikely to have occurred by chance alone.

No. The probability of obtaining this data is high enough to have been a chance occurrence.

Consider flipping nn times a coin. The probability for heads is given by pp where pp is some parameter which can be chosen from the interval (0,1)(0,1).

Write a Python code to simulate nn coin flips with heads probability pp and compute the running proportion of heads X¯nX¯n for nn running from 1 to 1,000 trials. Plot your results. Your plot should illustrate how the proportion of heads appears to converge to pp as nn approaches 1,000.

In [ ]:

### Insert your code here for simulating the coin flips and for computing the average

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In [2]:

### Complete the plot commands accordingly for also plotting the computed running averages in the graph below

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p = 0.25 # just an example

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plt.figure(figsize=(10,5))

plt.title("Proportion of heads in 1,000 coin flips")

plt.plot(np.arange(1000),p*np.ones(1000),'-',color="red",label="true probability") 

plt.xlabel("Number of coin flips")

plt.ylabel("Running average")

plt.legend(loc="upper right")

St. Andrew’s University receives 900 applications annually from prospective students. The application forms contain a variety of information including the individual’s scholastic aptitude test (SAT) score and whether or not the individual desires on-campus

housing.

What is the probability that a simple random sample of 30 applicants will provide an estimate of the population mean SAT score that is within plus or minus 10 (within 10 points) of the actual population mean m ? Given that the population Standard Deviation is 80.

Data show 648 applicants wanting On-Campus Housing. What is the probability that sample proportion exceeds 50%, when n =30?

What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicants desiring on-campus housing that is within plus or minus .05 of the actual population proportion?

If the University can provide to no more than 45% for the on –campus housing facilities, what would be the estimated number of accepted applicants desiring on-campus housing?

The mean systolic blood pressure for people in the United States is reported to be 122 millimeters of mercury (mmHg) with a standard deviation of 22.8 millimeters of mercury. The wellness department of a large corporation is investigating if the mean systolic blood pressure is different from the national mean. A random sample of 200 employees in a company were selected and found to have an average systolic blood pressure of 124.7 mmHg.

a) What is the probability a random employees blood pressure is higher than 135 mmHg.

b) What is the probability that 200 randomly selected employees mean blood pressure is greater than 127 mmHg.

c) The wellness department is providing a new health program to their employees. Past studies have shown 9.9 % of their employees have high blood pressure. Find the probability that if the wellness department examines 200 randomly selected employees, less than 12 employees will have high systolic blood pressure. Do you think the new program significantly lowers the number of employees with a high blood pressure.

Write a java program that simulates thousands of games and then calculate the probabilities from the simulation results. Specifically in the game, throw two dice, the possible summations of the results are: 2, 3, ..., 12. You need to use arrays to count the occurrence and store the probabilities of all possible summations. Try to simulate rolling two dice 100, 1000, 10,0000 times, or more if needed. Choose one simulation number so that the probabilities you calculated is within 1% absolute error compared with the theoretical probability. For example, the theoretical probability for summation result 2 is approximately 2.78%, then your calculated probability based on simulations should be between 2.28% and 3.28%. The following is required:

• Use an array of ints of length 11 to keep count of the difference occurrences. When an int or double array is created, its values are initialized to 0 by default.

• In the simulation loop, throw the two dice, add the values, and then increase the corresponding array element by 1.

• Turn the occurrence counts into probabilities after the simulation loop is done.