Tyrone paid $234.33 for a new watch, after tax, and the tax rate is 7%. What was the original price of the watch?
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Let s be the speed of the ship, and c be the speed of the current.
We know that distance equals speed multiplied by the time.
With the current
or
This is our first equation.
Now, against the current, we have
or
This is our second equation.
We solve the equations simultaneously, by adding them together
Substitute s=87.5 to the first equation to solve for c.
The speed of the ship is 87.5 km/hr and the speed of the current is 17.5 km/hr.
We know that distance equals speed multiplied by the time.
With the current
or
This is our first equation.
Now, against the current, we have
or
This is our second equation.
We solve the equations simultaneously, by adding them together
Substitute s=87.5 to the first equation to solve for c.
The speed of the ship is 87.5 km/hr and the speed of the current is 17.5 km/hr.
Answer:
a. 0.3372 or 33.72%; b. 0.2236 or 22.36%; c. 0.2879 or 28.79%; d. $112,351.00
Step-by-step explanation:
We need to use here the values from the <em>cumulative standard normal table</em> and <em>z-scores</em> to solve the questions. That is, all the values are transformed to a <em>z-score</em> to use the <em>standard normal table</em> to find the probabilities. Notice that the question is telling us about the <em>median</em> and not <em>mean</em>. Fortunately, in the normal distribution the mean, the median, and the mode are the same. So, we can say that:
As a result, the parameters for the normal distribution in this case are:
Then we can solve the questions as follows:
<h3>Part a: Probability that annual income of 90,000 or more</h3>We need to calculate the z-score of such a value of x=90,000:
We need to round this value to z = 0.42 to use the <em>cumulative standard normal table. </em>This value is above the mean (positive) and corresponds, approximately, with a cumulative probability of P(x<90000) = 0.66276.
Then, the probability that a household in Maryland has an annual income of $90,000 or more is:
Rounding to four decimal places is 0.3372 or 33.72%
<em>We can follow the same procedure to find the rest of the probabilities asked.</em>
<h3>Part b: Probability a household has an annual income of 50,000 or less.</h3>Rounding this value to two decimals (z = -0.76), we can conclude that this value is below the mean. To find this probability from the cumulative standard normal table, we first found the value of z = 0.76 (since no negative value is displayed in this table) and then subtracting this value from one. This is possible because the normal distribution is symmetrical. Then,
Thus, the probability that a household in Maryland has an annual income of $50,000 or less is 0.2236 (rounding to four decimals) or 22.36%.
<h3>Part c: Annual income between $40,000 and $70,000</h3>Mathematically, it can be expressed as:
For x = 40000:
<em>This value is below the mean and is -1.06085 standard deviations from it.</em>
Following the same procedure in Part b, the value for z = -1.06 corresponds to a cumulative probability of (P(z<1.06) = 0.85543):
For x = 70000:
Which corresponds to a cumulative probability of (P(z<0.17) = 0.56749):
Then, the probability that a household in Maryland has an annual income between $40,000 and $70,000 is:
0.43251 - 0.14457 = 0.28794.
Rounding to four decimals is 0.2879 or 28.79%.
Part d: Eighty-sixth percentile
The z-score that corresponds to a probability of 86% or 0.86 is z = 1.08.
Then, solving the equation for the corresponding z-score:
Then, the annual income of a household in the eighty-six percentile of annual household in Maryland is $112,351.00.