4,6,9,... Find the 7th term.
1 answer:
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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.