Suppose a carnival director in a certain city imposes a height limit on an amusement park ride called Terror Mountain, due to safety concerns. Patrons must be at least 4 feet tall to ride Terror Mountain. Suppose patrons’ heights in this city follow a Normal distribution with a mean of 4.5 feet and a standard deviation of 0.8 feet (patrons are mostly children). Make sure to show all of your work in this question. Show the distribution that your random variable follows; state the probability you are asked to calculate; show any tricks you use; show how you standardize, and state your found value from Table A4.

a) [5 marks] What is the probability that a randomly selected patron would be tall enough to ride Terror Mountain?

b) [5 marks] A group of 3 friends want to ride Terror Mountain. What is the probability that their mean height is greater than 4.5 feet?

c) [7 marks] Another group of 5 friends wants to ride Terror Mountain. What is the probability that their mean height is between 4 and 4.25 feet, inclusive?

In the probability distribution to the​ right, the random variable X represents the number of hits a baseball player obtained in a game over the course of a season. Complete parts​ (a) through​ (f) below. x ​P(x) 0 0.1685 1 0.3358 2 0.2828 3 0.1501 4 0.0374 5 0.0254 ​

(a) Verify that this is a discrete probability distribution. This is a discrete probability distribution because all of the probabilities are at least one of the probabilities is all of the probabilities are between 0 and 1​, ​inclusive, and the sum mean sum product of the probabilities is 1. ​(Type whole numbers. Use ascending​ order.)

​(b) Draw a graph of the probability distribution. Describe the shape of the distribution. Graph the probability distribution. Choose the correct graph below. A. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 Number of Hits Probability The graph of a probability distribution has a horizontal x-axis labeled "Number of Hits" from 0 to 5 in intervals of 1 and a vertical y-axis labeled "Probability" from 0 to 0.4 in intervals of 0.05. Vertical line segments are centered on each of the horizontal axis tick marks. The approximate heights of the vertical line segments are as follows, with the horizontal coordinate listed first and the line height listed second: 0, 0.15; 1, 0.04; 2, 0.03; 3, 0.17; 4, 0.34; 5, 0.28. B. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 Number of Hits Probability The graph of a probability distribution has a horizontal x-axis labeled "Number of Hits" from 0 to 5 in intervals of 1 and a vertical y-axis labeled "Probability" from 0 to 0.4 in intervals of 0.05. Vertical line segments are centered on each of the horizontal axis tick marks. The approximate heights of the vertical line segments are as follows, with the horizontal coordinate listed first and the line height listed second: 0, 0.34; 1, 0.15; 2, 0.03; 3, 0.17; 4, 0.28; 5, 0.04. C. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 Number of Hits Probability The graph of a probability distribution has a horizontal x-axis labeled "Number of Hits" from 0 to 5 in intervals of 1 and a vertical y-axis labeled "Probability" from 0 to 0.4 in intervals of 0.05. Vertical line segments are centered on each of the horizontal axis tick marks. The approximate heights of the vertical line segments are as follows, with the horizontal coordinate listed first and the line height listed second: 0, 0.03; 1, 0.04; 2, 0.15; 3, 0.28; 4, 0.34; 5, 0.17. D. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 Number of Hits Probability The graph of a probability distribution has a horizontal x-axis labeled "Number of Hits" from 0 to 5 in intervals of 1 and a vertical y-axis labeled "Probability" from 0 to 0.4 in intervals of 0.05. Vertical line segments are centered on each of the horizontal axis tick marks. The approximate heights of the vertical line segments are as follows, with the horizontal coordinate listed first and the line height listed second: 0, 0.17; 1, 0.34; 2, 0.28; 3, 0.15; 4, 0.04; 5, 0.03. Describe the shape of the distribution. The distribution has one mode has one mode is multimodal is uniform is bimodal and is skewed right. roughly symmetric. skewed right. skewed left.

​(c) Compute and interpret the mean of the random variable X. mu Subscript xequals 0.1666 hits ​(Type an integer or a decimal. Do not​ round.) Which of the following interpretations of the mean is​ correct? A. In any number of​ games, one would expect the mean number of hits per game to be the mean of the random variable. B. Over the course of many​ games, one would expect the mean number of hits per game to be the mean of the random variable. C. The observed number of hits per game will be less than the mean number of hits per game for most games. D. The observed number of hits per game will be equal to the mean number of hits per game for most games. ​

Need help with (c) through (f) please!

(d) Compute the standard deviation of the random variable X. sigma Subscript xequals nothing hits ​(Round to three decimal places as​ needed.)

​(e) What is the probability that in a randomly selected​ game, the player got 2​ hits? nothing ​(Type an integer or a decimal. Do not​ round.)

​(f) What is the probability that in a randomly selected​ game, the player got more than 1​ hit? nothing ​(Type an integer or a decimal. Do not​ round.)

Refer to the Lincolnville School District bus data.

1. Refer to the maintenance cost variable. The mean maintenance cost for last year is $4,552 with a standard deviation of $2,332. Estimate the number of buses with a maintenace cost of more than $6,000. Compare that with the actual number. Create a frequency distribution of maintenance cost. Is the distribution normally distributed?

2. Refer to the variable on the number of miles driven since the lastm maintenance. The mean is 11,121 and the standard deviation is 617 miles. Estimate the number of buses traveling more than 11,500 miles since the last maintnance. Compare that number with the actual value. Create a frequency distribution of miles since maintenance cost. Is the distribution normally distributed?

- Factory supervisors’ salaries -> product cost or period cost and why?

- Speakers used in Sony home-theater systems -> variable or fixed cost and why?

- Insurance costs related to a Mary Kay Cosmetics' manufacturing plant -> variable or fixed cost and why?

1) Identical twins Anna and Hannah visit you at the optical clinic. Anna, whose eyes can easily focus on distant objects (her far point), is also able to focus on objects within 20 cm of her eyes (her near point). Assuming the diameter and, hence, the distance between the cornea and retina, of Anna's eye is 20 mm, what is the range (in diopters) of Anna's vision? The limits of this range correspond to the total refractive power of her eyes at their far point and and the refractive power at their near point.

a) from 50 to 50.5 diopters

b) from 50 to 55 diopters

c) from 50 to 60 diopters

d) from 0 to 5 diopters

2) Hannah's eyes have the same range as her sister's, with the same focal power for her cornea (50 diopters) and for her variable lens (5 diopters), but Hannah suffers from myopia. She cannot focus on any object that lies more than 0.7 meters from her eyes since they are slightly longer -- the cornea to retina distance is larger -- than her sister's eyes. Considering this new far point, what is the diameter of Hannah's eyes (in millimeters, to the nearest tenth of a millimeter) assuming Anna's eye diameter was ideally, again, 20.0 mm? Hint: the focal power of the cornea remains the same for Hannah as for Anna for focusing distant objects, but the farthest Hannah can see (object distance) changes from infinity to 0.7 meters.  

3) Assuming, instead, that the diameters of Hannah's myopic eyes were 20.4 mm, but, again, that Hannah's eyes share the same focal powers for her cornea and lens as Anna's, what would be Hannah's near point (to the nearest tenth of a cm, in cm) if Anna's, again, is 20 cm?

4) Now assuming Anna's far point was found to be 0.8 m (i.e., her eyes can't focus on any object more than 0.8 m away), what power corrective lenses would you prescribe to Hannah so that, when wearing these lenses, her visual range was the same as Anna's (from a near point of 20 cm to a far point of infinity? Give your answer in units of diopters, to the nearest tenth of a diopter, with the correct sign.

5) One treatment of cataracts is to surgically remove the variable lens of the eye. If we assume that the cornea's refractive power focuses objects at infinite distances onto the retina of a person who has had this surgery, what power correcting lenses would they need to be able to read text at a 21-cm near-point distance? Again, give your answer in units of diopters, to the nearest tenth of a diopter and with the correct sign.

Answer the following questions in order to approximate the value of sin(0.8).

Note that π/ 4 ≈ 0.785 radians.

(a) Do you have enough information to use right triangles to estimate sin(0.8)? Why?

(b) Estimate sin(0.8) using values of sin(x) that you know from part (2).

(c) Estimate sin(0.8) using the graph of y = sin(x) from part (3).

(d) Estimate sin(0.8) using the 5th degree Taylor polynomial for sin(x) at a = 0 (I will accept either the Taylor polynomial up to n = 5 or the Taylor polynomial up to the x^5 term). Find the error bound from the Alternating Series Estimation Theorem (using the next nonzero term of the Maclaurin series), and explain what it tells you about the actual value of sin(0.8).

1. A single community is comprised of just three voters i = {1, 2, 3}, each of whom have differing tastes for public parks, as described by the following three demand functions.

q₁ = 100 – 2p

q₂ = 110 – 2p

q₃ = 126 – 2p  

(a)        Public parks are a local public good. Assuming that the marginal cost to society,

mc^s, of providing each unit of park space is $90, what is the socially optimal quantity of parks? Provide a graph with your answer. Please show all of your work. 3pt

(b)        Assume that the price tag for each unit of park is split evenly across the community

members so that the marginal cost to each member is just $30. At this price, what is

each member’s optimal quantity of park space? Is there unanimity across the three individuals regarding the desired level of park space? 3pt

(c)        Using Lindahl pricing (aka Lindahl taxing), what price schedule would guarantee unanimous agreement across all three members and would also yield a socially optimal outcome? Please show your work. 3pt

2. Please refer back to Q1 when answering the following questions.

(a)        Draw each individual’s demand curve for park space. Then calculate each person’s

consumer surplus at each of the three optimal quantities. Please show your work. 3pt

                        Hint: remember that each person must pay $30 per unit of park space consumed.

(b)        Using the consumer surplus calculations from Q2(a), fill in the following table by

assigning a rank to each person’s park space options. 3pt    

(c)        In a political environment with direct democracy through majority rule, which of the

three park space alternatives will consistently win a series of pair-wise votes? 3pt

d) Is the winning option aligned with what would be predicted by the median voter

            theorem? Is this outcome socially optimal? Explain why or why not. 3pt