The balance in the savings account after three years would be $5,705.79 if an initial deposit of $5,000 is made into a savings account with an annual interest rate of 5% compounded quarterly.
Compound interest formula can be used to determine the balance in an account after a certain period of time.
The formula used to calculate compound interest is as follows; A = P(1 + r/n)^nt Where; A represents the final amount, P represents the initial amount invested, r represents the annual interest rate, t represents the number of years invested, and n represents the number of times the interest is compounded per year.
Assuming that an initial deposit of $5,000 is made into a savings account with an annual interest rate of 5% compounded quarterly for three years, we can use the compound interest formula to compute the balance in the account at the end of three years. The first step is to calculate the quarterly interest rate. This can be done by dividing the annual interest rate by 4 since there are four quarters in a year. Hence, the quarterly interest rate would be;
r = 5% / 4r = 0.0125
Next, we calculate the total number of compounding periods (n) for three years if the interest is compounded quarterly. Since there are four quarters in a year, the number of compounding periods would be; n = 4 x 3n = 12
Substituting the values obtained into the compound interest formula, we have;
A = P(1 + r/n)^ntA = $5,000(1 + 0.0125/12)^(12 x 3)A = $5,705.79.
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The darkness of the print is measured quantitatively using an index. If the index is greater than or
equal to 2.0 then the darkness is acceptable. Anything less than 2.0 means the print is too light and
not acceptable. Assume that the machines print at an average darkness of 2.2 with a standard
deviation of 0.20.
(a) What percentage of printing jobs will be acceptable? (4)
(b) If the mean cannot be adjusted, but the standard deviation can, what must be the new standard
deviation such that a minimum of 95% of jobs will be acceptable?
84.13% of the printing jobs will be acceptable.
The new standard deviation required to achieve a minimum of 95% of jobs acceptable is 0.121.
The darkness of the print is measured quantitatively using an index. If the index is greater than or equal to 2.0 then the darkness is acceptable. Anything less than 2.0 means the print is too light and not acceptable. The machines print at an average darkness of 2.2 with a standard deviation of 0.20.
The mean of the darkness of the print is µ = 2.2 and the standard deviation is σ = 0.20.Therefore, the z-score can be calculated as; `z = (x - µ) / σ`.The index required for acceptable prints is 2.0. Thus, the percentage of prints that are acceptable can be calculated as follows;P(X ≥ 2.0) = P((X - µ)/σ ≥ (2.0 - 2.2) / 0.20)P(Z ≥ -1) = 1 - P(Z < -1)Using the standard normal table, P(Z < -1) = 0.1587P(Z ≥ -1) = 1 - 0.1587= 0.8413.
To find the new standard deviation, we can use the z-score formula.z = (x - µ) / σz = (2.0 - 2.2) / σz = -1Therefore, P(X ≥ 2.0) = 0.95P(Z ≥ -1) = 0.95P(Z < -1) = 0.05Using the standard normal table, the z-score value of -1.645 corresponds to a cumulative probability of 0.05. Hence,z = (2.0 - 2.2) / σ = -1.645σ = (2.0 - 2.2) / -1.645= 0.121.
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