Algebra 2/Trigonometry Regents Review Probability and Statistics

Permutations - The number of arrangements Practice 1: A four-digit serial number is to be created from the digits 0 through 9. How many of these serial numbers can be created if 0 can not be the first digit, no digit may be repeated, and the last digit must be 5?

1) 448 2) 504 3) 2,240 4) 2,520 Practice 2: Which formula can be used to determine the total number of different eight-letter arrangements that can be formed using the letters in the word DEADLINE?

1) 8! 2) 8!4! 3) !!!!!!!

4) !!!!∙!!

Practice 3: Which expression represents the number of different 8-letter arrangements that can be made from the letters of the word "SAVANNAH" if each letter is used only once?

1) !!!!

2) !!!!!!

3) 8P5 4) 8!

Practice 4: The letters of any word can be rearranged. Carol believes that the number of different 9-letter arrangements of the word “TENNESSEE” is greater than the number of different 7-letter arrangements of the word “VERMONT.” Is she correct? Justify your answer.

Combinations - When order is not important. Practice 5: An algebra class of 21 students must send 5students to meet with the principal. How many different groups of 5 students could be formed from this class? Practice 6: A basketball squad has ten players. Which expression represents the number of five-player teams that can be made if John, the team captain, must be on every team?

1) 𝐶!"  ! 2) 𝐶!   ! 3) 𝑃!   ! 4) 𝑃!"

 !

Counting Principle - If there are a ways for one activity to occur, and b ways

for a second activity to occur, then there are a • b ways for both to occur. Practice 7: Megan decides to go out to eat. The menu at the restaurant has four appetizers, three soups, seven entrees, and five desserts. If Megan decides to order an appetizer or a soup, and one entree, and two different desserts, how many different choices can she make? Binomial Probability – The probability of an event can be expressed as a binomial probability if its outcomes can be broken down into two probabilities p and q, where p and q are complementary (i.e. p + q = 1). If p is the probability of success then the probability of r successes in n trials can be determined by:

𝐶!   ! ∙ 𝑝!(1 − 𝑝)!!! It’s actually a very simple and easy formula.

Practice 8: The probability of winning a game is 3/5, then the probability of winning exactly 3 games out of 4 played is

1) 27/125 2) 54/625 3) 216/625 4) 532/625

Practice 9: If the probability that Mike will successfully complete a foul shot is 4/5, what is the probability that he will successfully complete exactly three of his next four foul shots?

1) 64/625 3) 256/625 2) 192/625 4) 64/125

Practice 10: In basketball, Nicole makes 4 baskets for every 10 shots. If she takes 3 shots, what is the probability that exactly 2 of them will be baskets?

1) 0.288 3) 0.600 2) 0.432 4) 0.960

Practice 11: Mrs. Gruber gave her history class a multiple choice quiz containing five questions. A student must answer at least four questions correctly to pass. Greg decided to guess on every question. If each of the four possible answers to each question is equally likely to be chosen, what is the probability that Greg passed the quiz? The Binomial Theorem can be used to expand binomials. The formula is provided on the formula page. Binomial Theorem 𝑎 + 𝑏 ! = 𝐶!   !𝑎!𝑏! + 𝐶!   !𝑎!!!𝑏! + + 𝐶!   !𝑎!!!𝑏! +  …  + 𝐶!   !𝑎!𝑏!

𝑎 + 𝑏 ! =   𝐶!   !𝑎!!!𝑏!!

!!!

Practice 12: Write the binomial expansion of 2𝑥 − 1 ! as a polynomial in simplest form. Most of the time, the Regents will only want an individual term. This makes it very easy. But don’t forget to start counting at zero! Practice 13: What is the fifth term in the expansion of (𝑥 + 2)!?

Practice 14: What is the third term in the expansion of 𝑎 − 2𝑏 !? Practice 15: The last term in the expansion of 𝑥 + 3𝑦 !? Practice 16: What is the fourth term in the expansion of 𝑎 − 𝑏 !? Statistical Methods: There are 3 basic ways to gather data: surveys, observational studies and controlled experiments. Only an experiment can be used to draw any cause and effect conclusions.

The population is the group from which statistical inferences will be drawn. For example, if we want to do a study about New Yorkers views on Same Sex marriage then the people living in NY are the population. If we want to see how many mosquitoes have malaria then mosquitoes are the population.

• A survey gathers information by having subjects answer questions. • An observational study gathers information about a treatment on subjects but the researchers can’t control

who gets the treatment and other factors. • A controlled experiment is conducted by dividing subjects into one or more groups. A researcher then gives

some groups a treatment and then compares the results to a control group that receives a placebo. • A census is a survey given to the entire population.

Practice 17: Which task is not a component of an observational study?

1) The researcher decides who will make up the sample. 2) The researcher analyzes the data received from the sample. 3) The researcher gathers data from the sample, using surveys or taking measurements. 4) The researcher divides the sample into two groups, with one group acting as a control group.

Practice 18: Howard collected fish eggs from a pond behind his house so he could determine whether sunlight had an effect on how many of the eggs hatched. After he collected the eggs, he divided them into two tanks. He put both tanks outside near the pond, and he covered one of the tanks with a box to block out all sunlight. State whether Howard's investigation was an example of a controlled experiment, an observation, or a survey. Justify your response.

More Statistics and using the TI-83/84 Standard deviation and variance are measures of spread. A large standard deviation indicates that there is a relatively large distance between data points. Population standard deviation is denoted with a lower case sigma, σ . Variance is standard deviation squared, σ2. Sample standard deviation is denoted by s. To enter a frequency table into your calculator hit STAT ENTER. Then get your statistics by pressing 1-VarStats L1, L2 Practice 19: Tanner and Robbie discovered that the means of their grades for the first semester in Mrs. Merrell’s mathematics class are identical. They also noticed that the standard deviation of Tanner's scores is 20.7, while the standard deviation of Robbie's scores is 2.7. Which statement must be true?

1) In general, Robbie's grades are lower than Tanner's grades. 2) Robbie's grades are more consistent than Tanner's grades. 3) Robbie had more failing grades during the semester than Tanner had. 4) The median for Robbie's grades is lower than the median for Tanner's grades.

The notation 𝒙 is read as “x bar” and it is the mean of X.

Practice 20: An electronics company produces a headphone set that can be adjusted to accommodate different-sized heads. Research into the distance between the top of people’s heads and the top of their ears produced the following data, in inches:

4.5, 4.8, 6.2, 5.5, 5.6, 5.4, 5.8, 6.0, 5.8, 6.2, 4.6, 5.0, 5.4, 5.8

The company decides to design their headphones to accommodate three standard deviations from the mean. Find, to the nearest tenth, the mean, the standard deviation, and the range of distances that must be accommodated. Practice 21 Conant High School has 17 students on its championship bowling team. Each student bowled one game. The scores are listed in the accompanying table. Find, to the nearest tenth, the population standard deviation of these scores. How many of the scores fall within one standard deviation of the mean? Normally Distributed Data If the data we gathered using a survey, observation or experiment is approximately normally distributed then we can draw conclusions based on the characteristics of a normal curve.

Practice 22: What percent of normally distributed data is within

One standard deviation Two Standard Deviations Three Standard Deviations

Practice 23: On a standardized (normally distributed) test, a score of 86 falls exactly 1.5 standard deviations below the mean. If the standard deviation for the test is 2, what is the mean score for this test?

1) 84 2) 84.5 3) 87.5 4) 89

Practice 24: On a standardized examination, Laura received a score of 85, which was exactly 2 standard deviations above the mean. If the standard deviation for the examination is 4, what is the mean for this examination?

1) 93 2) 87 3) 83 4) 77 Practice 25: In the accompanying diagram, the shaded area represents approximately 95% of the scores on a standardized test. If these scores ranged from 78 to 92, which could be the standard deviation?

1) 3.5 2) 7.0 3) 14.0 4) 20.0 Practice 26: The scores on a test have a normal distribution. The mean of the scores is 40 and the standard deviation is 6. The probability that a score chosen at random lies between 34 and 46 is closest to

1) .34 2) .68 3) .95 4) .99 Practice 27: The average score for a Latin test is 77 and the standard deviation is 8. Which percent best represents the probability that any one student scored between 61 and 93 on the test?

1) 95% 2) 99.5% 3) 68% 4) 34% Calculator Tip: normalcdf and invNorm can eliminate your need for the table here is how each work. Both are found in the dist menu. normalcdf(lower,upper,mean,stddev) – Returns the area under the curve between lower and upper. invNorm(percentile,mean,stddev) – Returns the score at the given percentile. Practice 28: The amount of time that a teenager plays video games in any given week is normally distributed. If a teenager plays video games an average of 15 hours per week, with a standard deviation of 3 hours, what is the probability of a teenager playing video games between 15 and 18 hours a week? Practice 29: The scores on a 100 point exam are normally distributed with a mean of 80 and a standard deviation of 6. A student's score places him between the 69th and 70th percentile. Which of the following best represents his score?

1) 66 2) 81 3) 84 4) 86

Practice 30: In a standardized test with a normal distribution of scores, the mean is 63 and the standard deviation is 5. Which score can be expected to occur most often?

1) 45 2) 55 3) 65 4) 74

Regression on the TI83/84

Diagnostics On Your calculator is capable of displaying even more information than it currently does. If you need to determine the correlation coefficient (r) of a scatter plot then you must enable diagnostics on your calculator. This only has to be done once. Press 2nd

0 then scroll down to DiagnosticsOn. Press Enter twice. Entering Data into L1 and L2 Press STAT ENTER to enter the editor. The first two columns should be L1 and L2 (If not, see troubleshooting). Enter your x-values into L1 and your y-values into L2. It is important that there is an equal number of values in both lists. Display a Scatter Plot To display a scatter plot of the data in L1 and L2 you should press Y= then use the cursor keys to highlight Plot 1. Note: You can “fine tune” your stat plot by pressing 2nd

Y= but the standard settings work well for Regents exams. Pressing Zoom 9 will display your data in an appropriate window. Perform a Linear Regression Press STAT ð and choose LinReg(ax+b) by pressing Enter on it. If you press Enter again you will get the equation of the best fit line. The output is given as a template. The “a” value will correspond to slope and the “b” value will be your y-intercept, as usual.

The equation of the best fit line in the example to the right would be: y = 1.8 x + 32

(The formula to convert Celsius to Fahrenheit) *Optional - If you want the equation automatically stored in Y1 during the regression process you just need to enter Y1 on the same line as LinReg(ax+b) BEFORE you hit Enter You can find Y1 under the VARS > Y-Vars> Function menu.

Troubleshooting

Problem Solution

L1 and L2 are missing Press STAT 5 to setup your editor Scatterplot doesn’t show up Make sure your PLOT 1 is highlighted and you’ve done a Zoom 9

Dim Mistmatch There are a different number of values in L1 and L2 You should be able to perform a variety of different regressions…

Practice 31: A box containing 1,000 coins is shaken, and the coins are emptied onto a table. Only the coins that land heads up are returned to the box, and then the process is repeated. The accompanying table shows the number of trials and the number of coins returned to the box after each trial.

Write an exponential regression equation, rounding the calculated values to the nearest ten-thousandth. Use the equation to predict how many coins would be returned to the box after the eighth trial.

Correlation coefficient is always between -1 and 1. 0 would denote no linear correlation. Practice 32: What could be the approximate value of the correlation coefficient for the accompanying scatter plot?

1) −0.85 2) −0.16 3) 0.21 4) 0.90

Practice 33: A linear regression equation of best fit between a student’s attendance and the degree of success in school is h = 0.5x + 68.5. The correlation coefficient, r, for these data would be

1) 0 < r < 1 2) −1 < r < 0 3) r = 0 4) r = −1