CALCU-Question 1(10 marks)A retailer receives deli

Solution by Steps

step 1

Let

A

be the event that the delivery is from Company A, and

B

be the event that the delivery is from Company B. We are given that

P(A) = 0.70

and

P(B) = 0.30

step 2

Let

U

be the event that an insulator is unsatisfactory. We are given that

P(U|A) = 0.05

and

P(U|B) = 0.03

step 3

We need to find the probability that less than 7 out of 100 insulators are unsatisfactory. This can be approximated using the binomial distribution

step 4

The expected number of unsatisfactory insulators from Company A is

E_A = 100 \times 0.05 = 5

. The expected number of unsatisfactory insulators from Company B is

E_B = 100 \times 0.03 = 3

step 5

We use the normal approximation to the binomial distribution. For Company A, the standard deviation is

\sigma_A = \sqrt{100 \times 0.05 \times 0.95} \approx 2.18

. For Company B, the standard deviation is

\sigma_B = \sqrt{100 \times 0.03 \times 0.97} \approx 1.70

step 6

We convert the number of unsatisfactory insulators to a z-score. For Company A,

z_A = \frac{7 - 5}{2.18} \approx 0.92

. For Company B,

z_B = \frac{7 - 3}{1.70} \approx 2.35

step 7

Using the standard normal distribution table, we find P(Z < 0.92) \approx 0.82 and P(Z < 2.35) \approx 0.99

step 8

The probability of finding less than 7 unsatisfactory insulators from Company A is P(U < 7|A) \approx 0.82. The probability of finding less than 7 unsatisfactory insulators from Company B is P(U < 7|B) \approx 0.99

step 9

We use Bayes' theorem to find the posterior probabilities. For Company A, P(A|U < 7) = \frac{P(U < 7|A) \cdot P(A)}{P(U < 7)}. For Company B, P(B|U < 7) = \frac{P(U < 7|B) \cdot P(B)}{P(U < 7)}

step 10

The total probability P(U < 7) = P(U < 7|A) \cdot P(A) + P(U < 7|B) \cdot P(B) = 0.82 \cdot 0.70 + 0.99 \cdot 0.30 \approx 0.87

step 11

Therefore, P(A|U < 7) = \frac{0.82 \cdot 0.70}{0.87} \approx 0.66 and P(B|U < 7) = \frac{0.99 \cdot 0.30}{0.87} \approx 0.34

Answer

It is more likely that the delivery is from Company A with a probability of approximately 0.66.

Key Concept

Bayes' Theorem

Explanation

Bayes' Theorem allows us to update the probability of an event based on new evidence. In this case, we used it to determine the likelihood of the delivery being from Company A or B given the number of unsatisfactory insulators.

Solution by Steps

step 1

To solve part (a), we need to select stocks from two sectors and calculate the coefficient of variation for the variable Price. The coefficient of variation (CV) is given by the formula:

CV = \frac{\sigma}{\mu}

, where

\sigma

is the standard deviation and

\mu

is the mean

step 2

For part (b)(i)(I), we need to form an estimated simple linear regression equation. The regression equation is of the form

Y = a + bX

, where

Y

is the dependent variable (Price/Earnings),

X

is the independent variable,

a

is the intercept, and

b

is the slope coefficient

step 3

To interpret the slope coefficient

b

, it represents the change in the dependent variable

Y

for a one-unit change in the independent variable

X

step 4

For part (b)(i)(II), the correlation coefficient

r

measures the strength and direction of the linear relationship between the two variables. The coefficient of determination

R^2

indicates the proportion of the variance in the dependent variable that is predictable from the independent variable

step 5

For part (b)(i)(III), to test the significance of the correlation at the

1\%

level, we use the t-test for the correlation coefficient. The null hypothesis

H_0

is that there is no correlation (

\rho = 0

)

step 6

For part (b)(ii), to analyze the correlation between Earnings/Share and EBITDA, we calculate the correlation coefficient

r

and determine its significance

step 7

For part (b)(iii)(I), we form a multiple regression equation with all the independent variables. The equation is of the form

Y = a + b_1X_1 + b_2X_2 + \ldots + b_nX_n

, where

Y

is the dependent variable,

X_1, X_2, \ldots, X_n

are the independent variables, and

b_1, b_2, \ldots, b_n

are the slope coefficients

step 8

For part (b)(iii)(II), the coefficient of determination

R^2

for the multiple regression model indicates the proportion of the variance in the dependent variable that is predictable from the independent variables

step 9

For part (b)(iii)(III), to test the overall significance of the multiple regression model at the

5\%

level, we use the F-test. The null hypothesis

H_0

is that all the slope coefficients are equal to zero

step 10

For part (b)(iii)(IV), to identify significant independent variables at the

1\%

level, we look at the p-values for each slope coefficient. Variables with p-values less than

0.01

are considered significant

step 11

For part (b)(iii)(V), we construct scatter plots for the significant independent variables against Price/Earnings and compare them with the conclusions from part (IV)

Answer

The solution involves calculating the coefficient of variation, forming and interpreting regression equations, and testing the significance of correlations and regression models.

Key Concept

Regression Analysis

Explanation

Regression analysis is used to understand the relationship between dependent and independent variables, and to make predictions based on this relationship.

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