General 3/4 Questions | Davis Rippon

Questions

1. A prestigious university only accepts students who score above the 98th percentile on a standardized entrance exam. If the exam scores are normally distributed with a mean of 80 and a standard deviation of 5, what is the minimum score required for admission?

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2. A student's test score has been standardized to a z-score of -1.2. If the mean score for the test is 85 and the standard deviation is 15, what is the student's actual test score?

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3. A population of exam scores follows a normal distribution with a mean of 85 and a standard deviation of 10.

i. What percentage of students scored above 95?

ii. What percentage of students scored below 60?

iii. To qualify for a scholarship, a student must have a score in the top 1% of the population. What score would be necessary to qualify for this scholarship?

iv. A student has a standardised score of 1.8. What is their actual score?

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4. A company takes out a loan of $50,000 at an annual interest rate of 8%, compounded quarterly. If the loan is to be repaid in 20 quarters with 19 equal payments, followed by a final payment that is as close to the regular payment as possible, what is the regular quarterly payment, rounded to the nearest cent?

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5. A charitable organization invested $100,000 in an annuity to provide a monthly stipend to a deserving student. The annuity earns interest at a rate of 4.2% per annum, compounded monthly. After six months, the annuity's balance is $63,419.19. If the student decides to reduce their monthly stipend to $1,800 to extend the life of the annuity, how many months will they receive the reduced stipend before the annuity is depleted with a final, smaller payment?

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6. A student's final grade for a subject is determined by three components: assignments, quizzes, and a final exam. Each component is weighted as follows: assignments are worth 15%, quizzes are worth 25%, and the final exam is worth 60%. If a student scored the following marks for a subject:

Semester 1: assignments 85, quizzes 90, exam 78

Semester 2: assignments 80, quizzes 85, exam 92

Represent the student’s scores in a 2 × 3 matrix and the weightings in a 3 × 1 matrix. Then, calculate the student’s final grade for the subject in each semester.

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Answers and Explanations

Answer 1:

$x = 80 + (2.05)(5) = 90.25$

To find the minimum score required for admission, we need to find the z-score corresponding to the 98th percentile, which is approximately 2.05. We can then use the z-score formula to find the corresponding x-value.

Answer 2:

$85 + (-1.2) \times 15 = 85 - 18 = 67$

To find the actual score, we multiply the z-score by the standard deviation and add the result to the mean.

Answer 3:

i. $P(X > 95) = P(Z > 1) = 1 - P(Z \leq 1) = 1 - 0.8413 = 0.1587$ or 15.87%

ii. $P(X < 60) = P(Z < -2.5) = 0.0062$ or 0.62%

iii. To find the score in the top 1%, we need to find the z-score corresponding to the top 1% of the distribution. Using a z-table, we find that this z-score is approximately 2.33. Then, we can find the corresponding score: $X = \mu + z\sigma = 85 + 2.33 \times 10 = 108.3$

iv. $X = \mu + z\sigma = 85 + 1.8 \times 10 = 103$

These questions require the application of the normal distribution to solve problems involving percentages and scores. The z-table is used to find the probabilities and z-scores, and then the actual scores are calculated using the mean and standard deviation of the distribution.

Answer 4:

The regular quarterly payment is $2,923.49.

To find the regular quarterly payment, we can use the formula for the present value of an annuity:

$A = P \frac{r(1+r)^n}{(1+r)^n - 1}$

where:

$A$ is the regular payment
$P$ is the principal amount ($50,000)
$r$ is the quarterly interest rate ( $\frac{0.08}{4} = 0.02$ )
$n$ is the number of payments (19)

Rearranging the formula to solve for $A$ , we get:

$A = \frac{P r (1+r)^n}{(1+r)^n - 1}$

Substituting the values, we get:

$A = \frac{50000 \times 0.02 \times (1+0.02)^{19}}{(1+0.02)^{19} - 1} \approx 2923.49$

Answer 5:

The student will receive the reduced stipend for 23 months.

To solve this problem, we first need to find the original monthly payment using the formula for the future value of an annuity:

$FV = PV \times \left(1 + \frac{r}{n}\right)^{nt} - PMT \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}$

where:

$FV$ = future value
$PV$ = present value (initial investment)
$r$ = annual interest rate
$n$ = number of times interest is compounded per year
$t$ = time in years
$PMT$ = monthly payment

We can rearrange this formula to solve for $PMT$ . Once we have the original monthly payment, we can calculate the number of months the reduced stipend will last using the same formula, but this time solving for $t$ .

Answer 6:

The student’s scores can be represented in the following 2 × 3 matrix:

$\begin{bmatrix} 85 & 90 & 78 \\ 80 & 85 & 92 \end{bmatrix}$

The weightings can be represented in the following 3 × 1 matrix:

$\begin{bmatrix} 0.15 \ 0.25 \ 0.60 \end{bmatrix}$

To calculate the student’s final grade for each semester, we multiply the scores matrix by the weightings matrix:

Semester 1: $\begin{bmatrix} 85 & 90 & 78 \end{bmatrix} \times \begin{bmatrix} 0.15 \ 0.25 \ 0.60 \end{bmatrix}^{T}$ $= (85 \times 0.15) + (90 \times 0.25) + (78 \times 0.60) = 12.75 + 22.5 + 46.8 = 82.05$

Semester 2: $\begin{bmatrix} 80 & 85 & 92 \end{bmatrix} \begin{bmatrix} 0.15 \ 0.25 \ 0.60 \end{bmatrix}^{T}$ $= (80 \times 0.15) + (85 \times 0.25) + (92 \times 0.60) = 12 + 21.25 + 55.2 = 88.45$

The student’s scores and the weightings are represented in matrix form, and then the final grade for each semester is calculated by multiplying the scores matrix by the weightings matrix.