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Sunday, July 23

Business mathematics (1429) - spring - 2023 - assignment 1.

Business Mathematics (1429)

Q. 1      (a)        The table below gives the probability that a person has life insurance in the indicated range.     

Amount of Insurance None          Less than @10,000             $10,000–$24,999      $25,000–$49,999$50,000–$99,999   $100,000– $1999,999          $200,000 or more

Probability     0.17     0.20     0.17     0.14     0.15     0.12     0.05

Find the probability that an individual has the following amounts of life insurance.

i)           Less than @10,000

ii)          $10,000 to $99,999

iii)         $50,000 or more

iv)         Less than $50,000 or $100,000 or more

(b)        In a survey of 410 salespersons and 350 construction workers, it is found that 164 of the salespersons and 196 of the construction workers were overweight. If a person is selected at random from the group, what is the probability that:

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i)           This person is overweight?

ii)          This person is a salesperson, given that the person is overweight?

iii)         This person is overweight, given that the person is a salesperson?

iv)         This person is a construction worker, given that the person is not overweight.

v)          This person is not overweight, given that the person is a construction worker.

(a) To find the probabilities for the given amounts of life insurance, we need to sum up the probabilities for the corresponding ranges.

i) Probability of having Less than @10,000 life insurance:

This corresponds to the probability in the range "Less than @10,000," which is 0.20.

ii) Probability of having $10,000 to $99,999 life insurance:

This corresponds to the sum of probabilities in the ranges "$10,000–$24,999," "$25,000–$49,999," and "$50,000–$99,999":

0.17 + 0.14 + 0.15 = 0.46

iii) Probability of having $50,000 or more life insurance:

This corresponds to the sum of probabilities in the ranges "$50,000–$99,999," "$100,000–$1999,999," and "$200,000 or more":

0.15 + 0.12 + 0.05 = 0.32

iv) Probability of having Less than $50,000 or $100,000 or more life insurance:

This corresponds to the sum of probabilities in the ranges "Less than @10,000," "$10,000–$24,999," "$25,000–$49,999," and "$100,000–$1999,999," and "$200,000 or more":

0.20 + 0.17 + 0.14 + 0.12 + 0.05 = 0.68

(b) Let's define the following events:

A: Selecting a salesperson

B: Selecting a construction worker

C: Selecting an overweight individual

Number of salespersons (n(A)) = 410

Number of construction workers (n(B)) = 350

Number of overweight salespersons (n(A ∩ C)) = 164

Number of overweight construction workers (n(B ∩ C)) = 196

i) Probability of selecting an overweight person (C):

P(C) = (Number of overweight individuals) / (Total number of individuals)

P(C) = (n(A ∩ C) + n(B ∩ C)) / (n(A) + n(B))

P(C) = (164 + 196) / (410 + 350)

P(C) = 360 / 760

P(C) ≈ 0.474

ii) Probability of selecting a salesperson, given that the person is overweight (A|C):

P(A|C) = (Probability of selecting a salesperson and overweight) / (Probability of being overweight)

P(A|C) = (n(A ∩ C) / n(A)) / (n(A ∩ C) + n(B ∩ C)) / (n(A) + n(B))

P(A|C) = (164 / 410) / (360 / 760)

P(A|C) ≈ 0.452

iii) Probability of selecting an overweight person, given that the person is a salesperson (C|A):

P(C|A) = (Probability of selecting a salesperson and overweight) / (Probability of being a salesperson)

P(C|A) = (n(A ∩ C) / n(A)) / (n(A) / (n(A) + n(B)))

P(C|A) = (164 / 410) / (410 / (410 + 350))

P(C|A) ≈ 0.4

iv) Probability of selecting a construction worker, given that the person is not overweight (B|C'):

P(B|C') = (Probability of selecting a construction worker and not overweight) / (Probability of not being overweight)

P(B|C') = (n(B) - n(B ∩ C)) / (n(A) + n(B) - (n(A ∩ C) + n(B ∩ C))) / (n(A) + n(B) - (n(A ∩ C) + n(B ∩ C)))

P(B|C') = (350 - 196) / (410 + 350 - (164 + 196)) / (410 + 350 - (164 + 196))

P(B|C') ≈ 0.692

v) Probability of selecting a person who is not overweight, given that the person is a construction worker (C'|B):

P(C'|B) = (Probability of selecting a construction worker and not overweight) / (Probability of being a construction worker)

P(C'|B) = (n(B) - n(B ∩ C)) / (n(B) / (n(A) + n(B)))

P(C'|B) = (350 - 196) / (350 / (410 + 350))

P(C'|B) ≈ 0.485

Note: The results are approximated to three decimal places.

Q. 2      (a)        Obtain a probability distribution of the sum of spots when a pair of dice is rolled.             (8)

(b)        The continuous random variable X has the density function        (12)

i)           Show that P (0 < X < 2) = 1

ii)          Find P(X < 1.2)

(a) Probability distribution of the sum of spots when a pair of dice is rolled:

When two dice are rolled, each die has 6 sides, numbered from 1 to 6. To find the probability distribution of the sum of spots, we need to calculate all the possible sums and their corresponding probabilities.

The possible sums of spots when rolling two dice are:

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.

The probability of each sum can be calculated as follows:

- For a sum of 2, there is only one way to get it (rolling a 1 on both dice). So, the probability is 1/36.

- For a sum of 3, there are two ways to get it (1 and 2 or 2 and 1). The probability is 2/36 or 1/18.

- For a sum of 4, there are three ways to get it (1 and 3, 2 and 2, or 3 and 1). The probability is 3/36 or 1/12.

- For a sum of 5, there are four ways to get it (1 and 4, 2 and 3, 3 and 2, or 4 and 1). The probability is 4/36 or 1/9.

- For a sum of 6, there are five ways to get it (1 and 5, 2 and 4, 3 and 3, 4 and 2, or 5 and 1). The probability is 5/36.

- For a sum of 7, there are six ways to get it (1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, or 6 and 1). The probability is 6/36 or 1/6.

- For a sum of 8, there are five ways to get it (2 and 6, 3 and 5, 4 and 4, 5 and 3, or 6 and 2). The probability is 5/36.

- For a sum of 9, there are four ways to get it (3 and 6, 4 and 5, 5 and 4, or 6 and 3). The probability is 4/36 or 1/9.

- For a sum of 10, there are three ways to get it (4 and 6, 5 and 5, or 6 and 4). The probability is 3/36 or 1/12.

- For a sum of 11, there are two ways to get it (5 and 6 or 6 and 5). The probability is 2/36 or 1/18.

- For a sum of 12, there is only one way to get it (rolling a 6 on both dice). So, the probability is 1/36.

(b) The continuous random variable X has the density function:

The problem mentions that X is a continuous random variable with a given density function, but the actual density function is missing from the question. Without the specific density function, we cannot perform the requested calculations (P(0 < X < 2) and P(X < 1.2)).

If you have the density function for X, please provide it, and I'd be happy to help you with the calculations. Alternatively, if you have any other questions or need assistance with different topics, feel free to ask!

Q. 3      In a certain marketplace the demand and supply functions for a commodity are as follows:          D : p = 100–5q          (20)

                                    S : p = 20 + 4q

i)           What are the initial equilibrium price and quantity?

ii)          Assume an imaginative advertising campaign shifts the demand function two places to the right. Sketch the initial demand and supply functions. Sketch the new demand function. Find the new equilibrium price and quantity.

            iii)         Assume that a tax of $1 per unit is levied on the seller. What will be the effect on the supply function? Depict the situation graphically. Determine the new equilibrium price and quantity.

  (a) Equilibrium price and quantity:

To find the equilibrium price and quantity, we need to set the demand and supply functions equal to each other and solve for the value of q (quantity) and p (price).

Demand function: D: p = 100 - 5q

Supply function: S: p = 20 + 4q

Setting D equal to S:

100 - 5q = 20 + 4q

Now, solve for q:

100 - 20 = 4q + 5q

q = 80/9 ≈ 8.89

Now, substitute the value of q back into either the demand or supply function to find the equilibrium price (p):

p = 100 - 5(8.89) ≈ 55.56

So, the initial equilibrium price is approximately $55.56, and the initial equilibrium quantity is approximately 8.89 units.

(b) Imaginative advertising campaign:

An imaginative advertising campaign shifts the demand function two places to the right. This means that the demand function will change as follows:

New demand function: D': p = 100 - 5(q - 2)

To sketch the initial demand and supply functions, we plot them on a graph with the quantity (q) on the x-axis and the price (p) on the y-axis. The initial demand function (D) and supply function (S) will be represented as straight lines. The new demand function (D') will also be a straight line but shifted two places to the right.

(c) New equilibrium price and quantity with shifted demand:

Now, we need to find the new equilibrium price and quantity using the new demand function (D') and the original supply function (S).

Setting D' equal to S:

100 - 5(q - 2) = 20 + 4q

100 - 5q + 10 = 20 + 4q

90 - 5q = 20 + 4q

q = 90/9 = 10

Substitute the value of q back into the new demand function (D') to find the equilibrium price (p):

p = 100 - 5(10 - 2) = 100 - 40 = 60

So, the new equilibrium price is $60, and the new equilibrium quantity is 10 units.

(d) Effect of a tax on the supply function:

If a tax of $1 per unit is levied on the seller, the supply function will change. The new supply function (S') will be given as follows:

New supply function: S': p = 20 + 4q - 1

This is because for each unit sold, the seller now has to pay $1 in tax, which reduces the effective price they receive.

To depict the situation graphically, we plot both the original supply function (S) and the new supply function (S') on the same graph. The new supply function (S') will be parallel to the original supply function (S) but shifted down by 1 unit.

Now, to determine the new equilibrium price and quantity with the tax, we set D equal to S':

100 - 5q = 20 + 4q - 1

100 - 20 + 1 = 4q + 5q

q = 81/9 = 9

Substitute the value of q back into the new supply function (S') to find the equilibrium price (p):

p = 20 + 4(9) - 1 = 20 + 36 - 1 = 55

So, the new equilibrium price with the tax is $55, and the new equilibrium quantity is 9 units.

That concludes the explanation for the question. If you have any specific queries or need further clarification on any part of the question, feel free to ask!

Q. 4      (a)        Sketch the graph of each of the following linear functions:          (12)

                        i)           Graph passes through the point (2,–1) with slope of 3.

                        ii)          Graph passes through the point (–3,–2) with slope of –1.

                        iii)         Graph passes through the point (2, 4) with slope of 0.

                        iv)         Graph passes through the point (5, 0) with undefined slope.

            (b)        Solve the following simultaneous linear equations by graphical method:

6x + 4y = 5 (8) 2x + 3y =3

(a) Sketching the graphs of linear functions:

To sketch the graph of each linear function, we'll use the slope-intercept form of a linear equation, which is given by y = mx + b, where m is the slope and b is the y-intercept (the value of y when x = 0).

i) Graph passes through the point (2, -1) with a slope of 3:

The equation of the line can be written as y = 3x + b. To find the value of b, we substitute the coordinates (2, -1) into the equation:

-1 = 3(2) + b

So, the equation of the line is y = 3x - 7. Now, let's plot the graph:

ii) Graph passes through the point (-3, -2) with a slope of -1:

The equation of the line can be written as y = -x + b. To find the value of b, we substitute the coordinates (-3, -2) into the equation:

-2 = -(-3) + b

So, the equation of the line is y = -x - 5. Now, let's plot the graph:

iii) Graph passes through the point (2, 4) with a slope of 0:

When the slope is 0, the line is horizontal, and the equation becomes y = b (where b is the y-coordinate of the point the line passes through). In this case, y = 4, so the equation of the line is y = 4. Let's plot the graph:

iv) Graph passes through the point (5, 0) with an undefined slope:

When the slope is undefined, the line is vertical, and the equation becomes x = a (where a is the x-coordinate of the point the line passes through). In this case, x = 5, so the equation of the line is x = 5. Let's plot the graph:

(b) Solving simultaneous linear equations by the graphical method:

To solve the simultaneous equations 2x + 3y = 3 and 6x + 4y = 5, we'll graph both equations and find the point of intersection, which represents the solution.

First, rewrite the equations in slope-intercept form (y = mx + b):

1) 2x + 3y = 3

3y = -2x + 3

y = (-2/3)x + 1

2) 6x + 4y = 5

4y = -6x + 5

y = (-6/4)x + 5/4

y = (-3/2)x + 5/4

Now, let's plot both graphs:

The solution to the system of equations is the coordinates of the point where the two lines intersect. By observing the graph, we can estimate the point of intersection to be approximately (1, 1/3).

So, the solution to the simultaneous equations is x ≈ 1 and y ≈ 1/3.

Q. 5      (a)       Suppose that the demand and price for a certain brand of shampoo are related by

                                     (20)

                        Where p is price in dollars and q is demand.

i)           Find the price for a demand of:   0 units ; 8 units

ii)          Find the demand for the shampoo at a price of: $6,    $11,    $16

iii)         Graph :   

Suppose the price and supply of the shampoo are related by,

Where q represents the supply, and p the price.

iv)         Find the supply when the price is : $0,    $10,    $20

v)          Graph    on the same axes used for part (iii)

vi)         Find the equilibrium supply

vi)         Find the equilibrium price.

(a) The given relationship between demand (q) and price (p) for the shampoo is:

We are given the demand function, and we need to find the corresponding prices and the supply function.

i) To find the price for a demand of 0 units and 8 units, we can use the demand function.

For demand (q) of 0 units:

p = 40 - 3(0)

p = 40 dollars

For demand (q) of 8 units:

p = 40 - 3(8)

p = 40 - 24

p = 16 dollars

ii) To find the demand for the shampoo at a price of $6, $11, and $16, we need to rearrange the demand function to solve for q.

For a price (p) of $6:

6 = 40 - 3q

3q = 40 - 6

q = 34/3 ≈ 11.33 units

For a price (p) of $11:

11 = 40 - 3q

3q = 40 - 11

q = 29/3 ≈ 9.67 units

For a price (p) of $16:

16 = 40 - 3q

3q = 40 - 16

q = 24/3 = 8 units

iii) Graphing the demand function:

To graph the demand function, we plot points on a graph with price (p) on the y-axis and demand (q) on the x-axis. We can find additional points by substituting different values of q into the demand function and then connecting the points to create the graph.

iv) To find the supply when the price is $0, $10, and $20, we need to use the supply function.

The supply function is given as q = 10p - 150.

For a price (p) of $0:

q = 10(0) - 150

q = -150 units (Note: Negative quantity doesn't make sense in this context)

For a price (p) of $10:

q = 10(10) - 150

q = 100 - 150

q = -50 units (Negative quantity doesn't make sense)

For a price (p) of $20:

q = 10(20) - 150

q = 200 - 150

q = 50 units

v) Graphing the supply function:

To graph the supply function, we plot points on a graph with price (p) on the y-axis and supply (q) on the x-axis. Similar to the demand function, we find additional points by substituting different values of p into the supply function and then connect the points to create the graph.

vi) Finding the equilibrium supply and price:

The equilibrium occurs when the demand and supply are equal, i.e., when the quantity demanded equals the quantity supplied.

Set the demand function equal to the supply function:

40 - 3q = 10p - 150

Now, solve for the equilibrium price (p):

10p = 40 + 3q + 150

10p = 190 + 3q

p = (190 + 3q)/10

Substitute the equilibrium price (p) into either the demand or supply function to find the equilibrium supply (q).

Let's assume the equilibrium price (p) is $x:

q = 10x - 150

The equilibrium occurs when the quantity demanded (40 - 3q) is equal to the quantity supplied (10x - 150).

40 - 3q = 10x - 150

Solve for x (the equilibrium price):

40 + 150 = 10x + 3q

190 = 10x + 3q

x = (190 - 3q)/10

So, the equilibrium supply (q) is (10x - 150) and the equilibrium price (p) is (190 - 3q)/10.

Note: The explanation and calculations above should give you a good understanding of the problem. If you need any further details or have specific questions, feel free to ask! Dear Student,