2.            The proportion of people who wait more than an hour at the Social Security Office is 28%. Use this information to answer the following questions:

A.            If you randomly select 45 people what is the probability that at least 34% of them will wait more than an hour?

B.            If you randomly select 60 people what is the probability that between 25% and 30% of them will wait more than an hour?

C.            If you randomly select 150 people what is the probability that less than 23% of them will wait more than an hour?

Let’s assume there is 1 fake coin out of 1000 coins. ( P[Coin=fake] = 0.001 ) The probability of showing head for fake coin is 0.9 (P[Head | Coin=fake] = 0.9). For normal coin, the probability of showing head is 0.5 (P[Head | Coin=normal] = 0.5).

i.  Bayes theorem, If you have a coin, and toss it one time. You got a head. What is the probability that this coin is fake?

ii. If you toss a coin 10 times, you got 8 head out of ten tosses. What is the probability of this event if the coin is fair (P[X=8|Coin=normal], X is a random variable representing number of head out of ten tosses)? What is the probability of this event if the coin is fake( P[X=8|Coin=fake])?

iii.   If you toss a coin 10 times, you got 8 head out of ten tosses. What is the probability that this coin is fake?

iv. Calculate the probability that you will get a head after you get 8 head out of ten tosses? What is the probability if you are Frequentist? What is it if you are Bayesian?

Suppose we know that a random variable X has a population mean µ = 400 with a standard deviation σ = 100. What are the following probabilities? (12 points)

The probability that the sample mean is above 376 when n = 1600.

The probability that the sample mean is above 376 when n = 400.

The probability that the sample mean is above 376 when n = 100.

The probability that the sample mean is above 376 when n = 64.

Let Y and Z be independent continuous random variables, both uniformly distributed between 0 and 1.

1. Find the CDF of |Y − Z|.
2. Find the PDF of |Y − Z|.

Mean=5.581 standard deviation= 1.114

Use 2 decimal places in this part and show all work leading to your answer. 1.) The IQR values are ___ to ___ 2.) IQR = __. 3.)The 15th percentile is __. 4.)The 85th percentile is __. 5.)Using complete sentences, and on your own paper, explain the meaning of the 15th and the 85th percentile of this distribution. 6.)The theoretical probability that a randomly chosen length is more than 6.5 cm =

A pet food company has a business objective of expanding its product line beyond its current kidney and shrimp-based cat foods. The company developed two new products, one based on chicken liver and the other based on salmon.  The company conducted an experiment to compare the two new products with its existing ones , as well as a generic beef-based product sold at a supermarket chain.

For the experiment, a sample of 35 cats from the population at a local animal shelter was selected. Seven cats were randomly assigned to each of the five products being tested.  Each of the cats was then presented with 3 ounces of the selected food in a dish at feeding time.  The researches defined the variable to be measured as the number of ounces of food that the cat consumed within a 10 minute time interval that began when the filled dish was presented.  The results of the experiment are summarized in the table below;

a)  State the appropriate null and alternative hypotheses for this experiment.

b) Use r coding to generate an ANOVAtable. Identify the Sum of Squares between (among) Groups, the Mean of Squares within Groups , the F statistic, and the p value.

c) Use your p value to determine if you are going to reject or fail to reject the null hypothesis at the .05 significance level.

d) Use the F statistic and the F critical value  F_c , to determine if you fail to reject the null hypothesis.  (Remember that the F_cvalue, and you reject  the null hypothesis if  F_stat> F_c.   You fail to reject the null hypothesis if  F_stat< F_c. )

QUESTION 23  Probability of Defective Products A bottling machine is designed to fill 16.75 fluid ounces of water in each water bottle. However, the actual fillings vary by small quantities. For many reasons, either too much water or too little water per filling is not desirable. Thus, the quality assurance manager puts out the following specification limits for the water content of each bottle: an LSL of 16.63 fluid ounces and a USL of 16.87 fluid ounces. Based on a very large sample of filled bottles, the manager sees that the fillings follow a normal distribution with µ = 16.78 fluid ounces and σ = 0.035 fluid ounces. (a)[2] Draw a normal distribution with specification limits to indicate the probabilities of defect. (b)[2] Calculate the probability of defect when too little water is filled in a bottle. (7 decimals) 3 (c)[2] Calculate the probability of defect when too much water is filled in a bottle. (7 decimals) (d)[1] Calculate the probability of defect when a bottle is filled. Calculate & interpret the DPMO. (e)[3] Calculate and make a statement about the sigma level of this bottling machine. Sketch the sigma level. (2 decimals)

A simple random sample of 80 items resulted in a sample mean of 60. The population standard deviation is σ = 5.

a.Compute the 95% confidence interval for the population mean. (Round your answers to two decimal places.)

____ to ___

b.Assume that the same sample mean was obtained from a sample of 160 items. Provide a 95% confidence interval for the population mean. (Round your answers to two decimal places.)

___ to ____

c.What is the effect of a larger sample size on the interval estimate?

A larger sample size provides a larger margin of error.

A larger sample size provides a smaller margin of error.    

A larger sample size does not change the margin of error.

The pressure of a pressure washer is normally distributed with a mean of 2510 psi and a standard deviation of 90 psi.

a) If all washers with pressure less than 2400 or greater than 2600 psi are considered defective, find the probability that a randomly selected washer is defective.

b) Determine the value of x, so that 99% of all washers have pressure within 2510 +/- x psi.

The operations manager of a large production plant would like to estimate the mean amount of time a worker takes to assemble a new electronic component. Assume that the standard deviation of this assembly time is 3.6 minutes.

After observing 120 workers assembling similar devices, the manager noticed that their average time was 16.2 minutes. Construct a 92% confidence interval for the mean assembly time.

How many workers should be involved in this study in order to have the mean assembly time estimated up to ±15 seconds with 92% confidence?

Design a correlational study, you will need two variables with at least five sets of data. between these two variables: time spent playing video games and aggression.

My question:

Assume the study produces a correlation of .56 between the variables. Analyze three possible causal reasons for the relationship.

The town of KnowWearSpatial, U.S.A. operates a rubbish waste disposal facility that is overloaded if its 5055 households discard waste with weights having a mean that exceeds 26.96 lb/wk. For many different weeks, it is found that the samples of 5055 households have weights that are normally distributed with a mean of 26.64 lb and a standard deviation of 12.56 lb.

What is the proportion of weeks in which the waste disposal facility is overloaded?
P(M > 26.96) =
Enter your answer as a number accurate to 4 decimal places. NOTE: Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Is this an acceptable level, or should action be taken to correct a problem of an overloaded system?

• No, this is not an acceptable level because it is not unusual for the system to be overloaded.
• Yes, this is an acceptable level because it is unusual for the system to be overloaded.

Statistics questions.

Given a small data set (n=5),

Estimate the population mean and its confidence interval for 99.997% confidence.

Estimate the population mean and its confidence interval for 95.00% and 80% confidence.

Estimate size of the precision intervals for each confidence level above.

Estimate the overall uncertainty for each confidence level above.

Use Excel to develop a regression model for the Hospital Database (using the “Excel Databases.xls” file on Blackboard) to predict the number of Personnel by the number of Births. Perform a test of the overall model, what is the value of the test statistic? Write your answer as a number, round your answer to 2 decimal places.