Researchers use Elisa to test for Lyme disease. Lyme disease infects 0.0009 of the US population each year. The test has a detection rate of 0.98 and a false positive rate of 0.2.

a. What is the probability of a positive test?

b. What is the probability that a person with a positive result actually has Lyme disease?

1. Determine statistically which two bands are the most correlated? Explain how did you get your answer and include intermediate calculations.

105 97 85 82 82 81 108 104 93 82 81 81 106 109 102 88 81 81 106 108 103 89 84 81 105 106 104 95 89 83 104 102 98 94 90 86 Green

128 115 95 89 89 90 129 124 109 94 89 89 128 133 125 102 93 89 129 134 124 101 95 90 130 128 126 112 104 92 128 125 118 108 104 96 Red

102 97 91 91 91 92 104 101 96 92 90 90 103 106 100 92 90 89 102 106 101 93 92 89 102 103 102 97 95 90 101 99 98 99 98 94 NIR

1. A company is to hire two employees. They have prepared a final list of eight candidates, all of whom are equally qualified. Of these eight candidates, five are women. If the company decides to select two persons randomly from these eight candidates, what is the probability that:

1. Both candidates are women.
2. The second candidate is a woman.
3. The first candidate is a woman given that second one is a woman.

1. A is called a palindrome if it reads the same from left and right. For instance, 13631 is a
palindrome, while 435734 is not. A 6-digit number n is randomly chosen. Find the probability
of the event that
(a) n is a palindrome.
(b) n is odd and a palindrome.
(c) n is even and a palindrome.

Section 1

Tennis players often spin a racquet to decide who serves first. Th e spun racquet can land with the manufacturer’s label facing up or down. A reasonable question to investigate is whether a spun tennis racquet is equally likely to land with the label facing up or down. (If the spun racquet is equally likely to land with the label facing in either direction, we say that the spinning process is fair.) Suppose that you gather data by spinning your tennis racquet 100 times, each time recording whether it lands with the label facing up or down.

1.1.1

a. Describe the relevant long-run proportion of interest in words.

b. What statistical term is given to the long-run proportion you described in (a)?

c. What value does the chance model assert for the long-run proportion?

d. Suppose that the spun racquet lands with the label facing up 48 times out of 100. Explain, as if to a friend who has not studied statistics, why this result does not constitute strong evidence against believing that the spinning process is fair.

e. Is the result in (d) statistically significant evidence that spinning is not fair or is it plausible that the spinning process is fair?

1.Assume that the heights of adult women are normally distributed with a mean height of 160 centimeters and the standard deviation is 8 centimeters.

What percentage of individuals have heights less than 160 centimeters?

2. Find the probability that a randomly selected individual has a height greater than 176 centimeters.

3. Find the probability that a randomly selected individual has a height less than 152 centimeters.

4. Can you determine the probability that a randomly selected individual has a height greater than 180 centimeters?

Question 3 [25]
OK furniture store submit weekly records the number of customer contacts contacted per week. A sample of 50 weekly reports showed a sample mean of 25 customer contacts per week. The sample standard deviation was 5.2. (Show all your works)
a) Compute the Margin of error at 0.05 significant level
[6]
b) Provide a 95% confidence interval for the population mean.
[4]
c) Compute the Margin of error at 0.01 significant level
[6]
d) Provide a 99% confidence interval for the population mean.
[4]
e) With a 0.99 probability, what size of sample should be taken if the desired margin of error is 1.5

Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $61, $99, and $135, respectively. The production requirements per unit are as follows:

	Number of Fans	Number of Cooling Coils	Manufacturing Time (hours)
Economy	1	1	8
Standard	1	2	12
Deluxe	1	4	14

For the coming production period, the company has 250 fan motors, 360 cooling coils, and 2600 hours of manufacturing time available. How many economy models (E), standard models (S), and deluxe models (D) should the company produce in order to maximize profit? The linear programming model for the problem is as follows:

Max	61E	+	99S	+	135D
s.t.
	1E	+	1S	+	1D	≤	250	Fan motors
	1E	+	2S	+	4D	≤	360	Cooling coils
	8E	+	12S	+	14D	≤	2600	Manufacturing time
E, S, D ≥ 0

The sensitivity report is shown in the figure below.

Optimal Objective Value =      19430.00000
		Variable		Value		Reduced Cost
		E		140.00000		0.00000
		S		110.00000		0.00000
		D		0.00000		40.00000
		Constraint		Slack/Surplus		Dual Value
		Fan motors		0.00000		23.00000
		Cooling coils		0.00000		38.00000
		Manufacturing time		160.00000		0.00000

		Variable		Objective Coefficient		Allowable Increase		Allowable Decrease
		E		61.00000		20.00000		11.50000
		S		99.00000		23.00000		13.33333
		D		135.00000		40.00000		Infinite
		Constraint		RHS Value		Allowable Increase		Allowable Decrease
		Fan motors		250.00000		40.00000		70.00000
		Cooling coils		360.00000		40.00000		110.00000
		Manufacturing time		2600.00000		Infinite		160.00000

1. Identify the range of optimality for each objective function coefficient. If there is no limit, then enter the text "NA" as your answer. If required, round your answers to one decimal place.
   

   	Objective Coefficient Range
   Variable	lower limit	upper limit
   E
   S
   D

2. Suppose the profit for the economy model (E) is increased by $6 per unit, the profit for the standard model (S) is decreased by $2 per unit, and the profit for the deluxe model (D) is increased by $4 per unit. What will the new optimal solution be? If required, round your answers to three decimal places. If your answer is zero, enter "0".
   

   Optimal Solution
   E
   S
   D

   
   If required, round your answer for Total Profit to two decimal places.
   
   Total Profit: $_____
   
3. Identify the range of feasibility for the right-hand-side values. If there is no limit, then enter the text "NA" as your answer. If required, round your answers to one decimal place.
   

   	Right-Hand-Side-Range
   Constraints	lower limit	upper limit
   Fan motors
   Cooling coils
   Manufacturing time

4. If the number of fan motors available for production is increased by 60, will the dual value for that constraint change?
   
   Yes  because the allowable increase for fan motors is_____without changing the optimal solution.

The following require calculating the probability of the specified event based on an assumed probability distribution. Remember to consider whether the event involves discrete or continuous variables.

You are measuring height of vegetation in a grassland using a Robel pole and a 5 m. radius. Based on 100 random samples from the grassland, you obtain a mean height of 0.6 m with a standard deviation of 0.04 m2.

a) What distribution is the appropriate reference for this problem?

b) Ninety percent of the samples are expected to be under what height? Use you will need to use the appropriate command in R of d<dist>, p<dist>, q<dist>, or r<dist> and use the appropriate values as arguments. Use help(command) to find out what these arguments are for your distribution, e.g., help(qbinom) will give you the help for this command.

A survey is taken among customers of a fast-food restaurant to determine preference for hamburger or chicken. Of 200 respondents selected, 75 were children and 125 were adults. 120 preferred hamburger and 80 preferred chickens. 55 of the children preferred hamburger and 20 preferred chickens. Set up a 2x2 contingency table using this information and answer the following questions:

Age/ Food	Hamburger	Chicken	Total
Child
Adult
Total			200

What is the probability that a randomly selected individual is an adult?

What is the probability that a randomly selected individual is a child and prefers chicken?

Given the person is a child, what is the probability that this child prefers a

hamburger?

Assume we know that a person has ordered chicken, what is the probability that this individual is an adult?

Are food preference and age statistically independent?

2) Three messenger services deliver to a small town in Oregon. Service A has 60% of all the scheduled deliveries, service B has 30%, and service C has the remaining 10%. Their on-time rates are 80%, 60%, and 40% respectively. Define event O as a service delivers a package on time.

Calculate P(A and O)

Calculate P(B and O)

Calculate P(C and O)

Calculate the probability that a package was delivered on time.

If a package was delivered on time, what is the probability that it was service A?

If a package was delivered 40 minutes late, what is the probability that it was service A?

3) The number of power outages at a nuclear power plant has a Poisson distribution with a mean of 6 outages per year.

What is the probability that there will be exactly 3 power outages in a year?

What is the probability that there will be at least 1 power outage in a year?

What is the variance for this distribution?

What is the mean power outage for this nuclear power plant in a decade?

           

           

Based on annual driving of 15,000 miles and fuel efficiency of 20 mpg, a car in the United States uses, on average, 700 gallons of gasoline per year. If annual automobile fuel usage is normally distributed, and if 26.76% of cars in the United States use less than 480 gallons of gasoline per year, what is the standard deviation?

Round your answer to 2 decimal places, the tolerance is +/-0.05.

Suppose that nn independent trials are performed, with trial ii being a success with probability 1/(2i+1).12i1. Let PnPndenote the probability that the total number of successes that result is an odd number.

1.Find Pn for n=1,2,3,4,5.

2.Conjecture a general formula for Pn.

3. Derive a formula for Pn in terms of Pn−1

Verify that your conjecture in part (b) satisfies the recursive formula in part (c). Because the recursive formula has a unique solution, this then proves that your conjecture is correct.

The manufacturer of a portable music player (PMP) has shown that the average life of the product is 72 months with a standard deviation of 12 months. The manufacturer is considering using a new parts supplier for th PMP's and want to test that the new hard drives will increase the life of the PMP. Before manufacturing the PMP's on a lareg scale, the manufactuer sampled 200 PMP's and found the average life to be 78 months. Test the hypothesis using alpha = .01 that the new hard drives will increase the life of the PMP's. Assume the standard deviation of the new PMP's is the same as the standard deviation of the older model.

A website developer has indicated to potential clients that for the sites he has developed visitors spend an average of 45 minutes per day on the sites. One of his potential clients conducted a survey of 20 visitors to several of his sites and found that the average time spent was 35 minutes with a standard deviation of 7 minutes. Determine if there is sufficient evidence to conclude that the average time spent on the sites is different from what he indicated. Conduct the test at the 0.05 level.

In both cases, in addition to testing the hypotheses using a critical value, also calculate the p value for the test statistic.

A random sample of 43 taxpayers claimed an average of $9,853 in medical expenses for the year. Assume the population standard deviation for these deductions was ​2,418. Construct confidence intervals to estimate the average deduction for the population with the levels of significance shown below.

a.1%

b.5%

c.20%

a. The confidence interval with a 1% level of significance has a lower limit of _____ and an upper limit of ______.

b. The confidence interval with a 5% level of significance has a lower limit of _____ and an upper limit of ______.

c. The confidence interval with a 20% level of significance has a lower limit of _____ and an upper limit of ______.

In looking at our class’s data as a sample of a larger population of students (who have taken, are taking, or may one day take this class), we find that the mean number of hours exercised per week during the summer is nearly 9 hours. We know that this is an estimate however. Is it likely that the true population mean is actually under 7 hours? Use a 95% confidence interval to determine this. If we’re willing to use a 99% confidence interval, does that change our findings? (Careful with your rounding!)

. mean exersum

Mean estimation                   Number of obs   =        215

--------------------------------------------------------------

             |       Mean   Std. Err.     [95% Conf. Interval]

-------------+------------------------------------------------

     exersum |   8.946512   .7143183