1) Sketch two normal distribution curves both with a standard deviation of \(2\) but one with a mean of \(10\) and the other with a mean of \(12\).
2) Assuming that the heights of adult men are normally distributed with a mean of \(70\) inches and a standard deviation of \(3.5\) inches, answer the following questions and sketch the distribution.
a) What is the percentage of men taller than \(63\) inches?
b) What is the percentage of men between \(63\) and \(77\) inches?
c) What is the percentage of men between \(70\) and \(73.5\) inches?
d) What heights describe the top \(16\%\) of men?
3) Women's \(100\)m dash times are normally distributed with a mean of \(34\) seconds and a standard deviation of approximately \(3\) seconds. Sketch the distribution.
If \(1000\) women run, how many women do we expect to finish in..
a) In \(31\) to \(37\) seconds?
b) In \(34\) to \(40\) seconds?
c) In \(40\) seconds or longer?
4) In a college class of \(120\), Leighton received a final exam grade of \(95\). The grades were distributed normally with a mean of \(74\) and standard deviation of \(6\). How many standard deviations away is Leighton’s grade from the mean? Would her score be considered an outlier?
5) The average car payment for residents in Naperville, IL is \(\$348\) per month with an standard deviation of \(\$20\). Sketch the distribution and give \(3\) conclusions that you can make about monthly car payments in Illinois.
For all problems that require a z-score conversion you may either use your table or calculator, note that answers may vary slightly in the answer bank because of this but should not be dramatically different.
6) Sketch the areas under the standard normal curve over the indicated intervals and find the probabilities.
a) \(P(z \geq 1.45)\)
b) \(P(z < 0.58)\)
c) \(P( -1 < z \leq 1)\)
d) \(P( -4 <z < 4)\)
7) On an actuarial exam, the mean score is \(72\) points and the standard deviation is \(11\) points. What is the probability that a randomly selected score is between \(72\) and \(80\)?
8) The average daytime temperature in Orlando, Florida in the month of July is \(92 ^{\circ}\) with a standard deviation of \(4 ^{\circ}\). If you head out to a theme park on a random day in July, find...
a) The probability that the temperature will be higher than \(85 ^{\circ}\).
b) The probability that the temperature will be in the \(90\)s.
c) The probability that the temperature will be below \(95^ {\circ}\).
9) The average salary for a math professor with a PhD at a four year university is \(\$77,000\) with a standard deviation of \(\$8,400\). If such a professor is randomly selected, find...
a) The probability that they make more than \(\$80,000\).
b) The probability that they make between \(\$60,000\) and \(80,000\).
10) Explore the following graph and calculations. If \(a=-1\) and \(b=1\), what decimal answer does the third line give? If \(a=-2\) and \(b=2\), what decimal answer does the third line give? How are these numbers related to what we have learned this chapter? You may want to research the "elongated s" symbol online to discover what is being calculated.
2) Assuming that the heights of adult men are normally distributed with a mean of \(70\) inches and a standard deviation of \(3.5\) inches, answer the following questions and sketch the distribution.
a) What is the percentage of men taller than \(63\) inches?
b) What is the percentage of men between \(63\) and \(77\) inches?
c) What is the percentage of men between \(70\) and \(73.5\) inches?
d) What heights describe the top \(16\%\) of men?
3) Women's \(100\)m dash times are normally distributed with a mean of \(34\) seconds and a standard deviation of approximately \(3\) seconds. Sketch the distribution.
If \(1000\) women run, how many women do we expect to finish in..
a) In \(31\) to \(37\) seconds?
b) In \(34\) to \(40\) seconds?
c) In \(40\) seconds or longer?
4) In a college class of \(120\), Leighton received a final exam grade of \(95\). The grades were distributed normally with a mean of \(74\) and standard deviation of \(6\). How many standard deviations away is Leighton’s grade from the mean? Would her score be considered an outlier?
5) The average car payment for residents in Naperville, IL is \(\$348\) per month with an standard deviation of \(\$20\). Sketch the distribution and give \(3\) conclusions that you can make about monthly car payments in Illinois.
For all problems that require a z-score conversion you may either use your table or calculator, note that answers may vary slightly in the answer bank because of this but should not be dramatically different.
6) Sketch the areas under the standard normal curve over the indicated intervals and find the probabilities.
a) \(P(z \geq 1.45)\)
b) \(P(z < 0.58)\)
c) \(P( -1 < z \leq 1)\)
d) \(P( -4 <z < 4)\)
7) On an actuarial exam, the mean score is \(72\) points and the standard deviation is \(11\) points. What is the probability that a randomly selected score is between \(72\) and \(80\)?
8) The average daytime temperature in Orlando, Florida in the month of July is \(92 ^{\circ}\) with a standard deviation of \(4 ^{\circ}\). If you head out to a theme park on a random day in July, find...
a) The probability that the temperature will be higher than \(85 ^{\circ}\).
b) The probability that the temperature will be in the \(90\)s.
c) The probability that the temperature will be below \(95^ {\circ}\).
9) The average salary for a math professor with a PhD at a four year university is \(\$77,000\) with a standard deviation of \(\$8,400\). If such a professor is randomly selected, find...
a) The probability that they make more than \(\$80,000\).
b) The probability that they make between \(\$60,000\) and \(80,000\).
10) Explore the following graph and calculations. If \(a=-1\) and \(b=1\), what decimal answer does the third line give? If \(a=-2\) and \(b=2\), what decimal answer does the third line give? How are these numbers related to what we have learned this chapter? You may want to research the "elongated s" symbol online to discover what is being calculated.
11) The average birth weight for a baby in the United States is \(7.5\) pounds and the standard deviation is \(1.25\) pounds. If a baby is born in the \(85\)th percentile for weight, how much do they weigh?
12) The scores of SAT tests are normally distributed with an average score of \(1020\) and a standard deviation of \(194\) points. Find the actual SAT score for the following percentiles:
a) the \(90\)th percentile
b) the \(70\)th percentile
c) the \(45\)th percentile
d) the range for the lowest \(20\%\) of scores (the minimum score on a part of the SAT is \(400\))
e) the range for the middle \(50\%\) of scores
f) the range for the top \(5\%\) of scores (the maximum score on a part of the SAT is \(1600\))
13) The scores for a Geometry test are normally distributed with an average of \(70\%\) and a standard deviation of \(11\%\). Find the cutoff scores for the following breakdown of grades.
\(15\%\) are A’s
\(25\%\) are B’s
\(40\%\) are C’s
\(15\%\) are D’s
Review
14) Solve the following without a calculator \(4^{2x}=8^{x-1}\).
15) If \(\cos (\theta) >0\) and \(\tan (\theta) >0\) which quadrant is the terminal side of \(\theta\) located in?
16) Evaluate the following without a calculator \(log_{\frac{1}{2}} 8\).
17) What is the difference between a function and a sequence?
Solution Bank