stats homework 3

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California State University, Long Beach *

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2197

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Computer Science

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Feb 20, 2024

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Problem 1 Part (a): Sample Points The sample space for drawing two marbles without replacement from a box containing two blue marbles and three red marbles consists of the following outcomes: - BB (Both Blue) - BR (First Blue, Second Red) - RB (First Red, Second Blue) - RR (Both Red) Part (b): Assign Probabilities to the Sample Points The probabilities for each of the outcomes are calculated as follows: - P(BB) (Probability of drawing two blue marbles): 0.1 or 10% - P(BR or RB) (Probability of drawing a blue and a red marble, in any order): 0.6 or 60% - P(RR) (Probability of drawing two red marbles): 0.3 or 30% These probabilities are based on combinatorial calculations considering the total number of ways to draw 2 marbles out of 5 without replacement. The total number of such combinations is 10. Part (c): Probability of Observing Each Event - Event A (Two blue marbles are drawn): The probability is 0.1 or 10%. - Event B (A red and a blue marble are drawn): The probability, considering both orders (BR and RB), is 0.6 or 60%. - Event C (Two red marbles are drawn): The probability is 0.3 or 30%. Problem 2 Part (a): Total Number of License Plates The total number of license plates that can be made with three letters followed by three digits is 17,576,000. Part (b): License Plates Without "Q" and "9" The total number of license plates that contain neither the letter "Q" nor the digit "9" is 11,390,625. Part (c): Probability of Drawing a Plate Without "Q" and "9" The probability that a randomly drawn license plate contains neither the letter "Q" nor the digit "9" is approximately 0.648 or 64.8%. Problem 3 Part (a): Assignments to Shifts The company can assign the 15 new employees to the day shift, graveyard shift, and night shift in 630,630 di ff erent ways . Part (b): Probability of Matching Socks from Two Drawers • The probability of drawing a blue sock from each drawer is approximately 0.381 . • The probability of drawing a white sock from each drawer is approximately 0.143 . • Therefore, the total probability of drawing matching socks (either both blue or both white) is approximately 0.524 .

Part (c): Probability of Matching Socks from One Drawer The probability of drawing two matching socks (either both red, both green, or both black) from the drawer is approximately 0.333 , or one-third. Problem 4 Part(a): The probability that a randomly selected mouse was species 1 is approximately 49.54% . Part(b): The probability that a randomly selected mouse was infected is approximately 51.38% . Part(c): The probability that a randomly selected mouse was both infected and species 1 is approximately 33.03% . Part(d): The probability that a randomly selected mouse was not infected and species 2 is approximately 32.11% . Problem 5 Part (a): The estimated probability that a mouse was infected given it is species 1 is approximately 66.67% . Part (b): The estimated probability that a mouse was infected given it is species 2 is approximately 36.36% . Part (c): The estimated probability that an infected mouse was species 1 is approximately 64.29% . Part (d): To determine if the events of a mouse being species 1 and a mouse being infected are independent, we compare the probability of a mouse being infected given it is species 1 or 2 to the overall probability of being infected.The overall probability of a mouse being infected is 51.38% . Since the probability of being infected given the mouse is species 1 ( 66.67% ) and species 2 ( 36.36% ) are di ff erent from the overall probability of being infected, the events are not independent . The di ff erence in probabilities indicates a dependency between the species of the mouse and the likelihood of infection Problem 6 Part(a): The probability that the target is hit would be 0.65 as I would do 1-P(both miss) Part(b): The probability that the target is hit by exactly one shot would be .50 Part(c): Since we want exactly one shot, and we need it to be Laura’s it would be .7. Problem 7 Part(a) The probability that the customer is a good risk and has filed a claim is approximately 0.0035 or 0.35% . Part(b) The probability that the customer has filed a claim is 0.008 or 0.8% .

Part(c) Given that the customer has filed a claim, the probability that the customer is a good risk is approximately 0.4375 or 43.75% . Problem 8 Part(a): False because the intersection of two events represents all the outcomes that both A and B would have. The Union of the two events would represent the set of outcomes that are in either A or B or both. Since the intersection would never have more outcomes than the possible outcomes, this is false. Part(b): True because the probability of A will be smaller than or equal to the union of two events because this will be all the events that A and B occur. This means that it will account for the probability of A as well as adding whatever additional outcomes that B will have. Part(c): False because The conditional probability of A given B (denoted as P(A ∣ B)is the probability that event A occurs given that B has already occurred. This concept is di ff erent from the intersection A ∩ B, which is a set of outcomes. The comparison in the statement is not valid because P(A ∣ B) is a measure of likelihood given a condition, while A ∩ B refers to a set of outcomes. Furthermore, conditional probability values range between 0 and 1 and do not directly compare in size to a set of outcomes. R Homework Problem I A.) B.)

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Problem II A.) The issue with this code is that it attempts to map the aesthetic ‘color’ to a fixed value "blue" within the aes() function, which is meant for mapping data to aesthetics dynamically. To make all points blue, the color should be set outside of ‘aes()’ B.) When mapping a continuous variable to color, ggplot2 will use a gradient scale. For categorical variables, it will use distinct colors for each category. Mapping a continuous variable to size will vary the size of the points continuously. For categorical variables, size di ff erences are less meaningful. Shape is typically used for categorical variables since there are a limited number of shapes available. Mapping a continuous variable to shape is not standard practice and can lead to unclear visualizations. C.) When the same variable is mapped to multiple aesthetics, each aesthetic will represent the variable's values accordingly. This could make the plot more informative or cluttered, depending on the context and how the aesthetics are interpreted together. D.) The stroke aesthetic in ggplot2 controls the size of the border around shapes that have a border, like points of certain shapes. It works with shapes that are not filled or have borders. E.) Mapping an aesthetic to a logical expression like dispel < 5 will divide the data into two groups based on whether the condition is ‘TRUE’ or ‘FALSE’. Points will be colored di ff erently based on the outcome of this condition, e ff ectively creating a binary color mapping based on the specified threshold.

stats homework 3

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