Algebra 2/Trig: Chapter 15 Statistics In this unit, we will Find sums expressed in summation notation Determine measures of central tendency Use a normal distribution curve to determine theoretical percentages of an event occurring Determine best-fit equations to model data and use those equations to make predictions on future events
A2T Chapter 15: Lesson #1 - Summation Notation In mathematics, there are often times where large sums of numbers need to be added, usually according to a certain pattern. Instead of writing out these sums longhand, we use summation notation, which is represented by the Greek capital letter for S, which is called sigma. (It s a big S folks, not the big E as everyone seems to like to call it!!) Multiplier. Sometimes absent. When finished, multiply your final answer by this constant. 1 2 5 2k i 1 Dummy variable. This serves as a counter. Whatever it = is the initial value. Concept 1: Finding the Sum of a Series 2 Ending value of the dummy variable. Expression in terms of the dummy variable. May contain other variables, too.
Regents Question:
Concept 2: Writing a Sequences in Summation Notation You try it! Regents Question:
Working Backwards SUMMARY Exit Ticket
Lesson #1 HW: Summation Notation
Answer Key:
Algebra2/Trig: Central Tendency and the NORMAL Distribution (Chapters 15-2 15-6) You ve already met the Measures of Central Tendency in 9 th grade algebra: Mean: The Arithmetic mean, where you add up all the data and divide by the number of data items. From now on, the will be called x-bar and will be written like this:. Median: The middle number, if all of the data are lined up in order. If there are two data items in the middle, the average of the two is taken. Mode: The data item that occurs the most. Although we usually use the mean in a school setting, the median is used quite frequently, as it doesn t take into consideration data that is either much higher than the rest, or much lower than the rest. Measures of center do not tell the whole story of what the data is like. There are also measures of spread. How consistent or spread out the data is. You ve also met some of these: The quartiles and the range. Range: Maximum value minimum value Q1: The 25% percentile: The value at which the lower 25% of data lies. The median: The 50% percentile: The value at which half of the data lies below. Q3: The 75% percentile: The value at which the lower 75% of data lies. Your calculator will do all of this work for you. To enter lists of data To calculate the statistics for A LIST data Make sure the list is in L1 ONLY To calculate the statistics for A TABLE of data Make sure the data item is in L1 and the FREQUENCY is in L2 1. 2. Enter the data in L1 and/or L2 3. Go back to the home screen Calculator should say 1-VAR STATS Calculator should say 1-VAR STATS L1, L2
The Normal Distribution If many measurements of a specific trait of a specific population are taken, the distribution is (often) an approximate bell curve, or normal distribution. For example, the following groups of data should all be approximately normal: Heights of American females Weights of French males Lengths of salmon found in a certain river in Alaska The sums of two dice thrown a multitude of times Lots of measurements cannot be modeled by a normal curve. For example, wait time for people standing on the platform of the NYC-bound Metro-North train station In a theoretical normal distribution: The curve is perfectly symmetric The mean and the median are the same. Most of the measurements will be very close to the mean. Very few of the measurements will be very far from the mean. Normal curves, taken from http://en.wikipedia.org/wi ki/normal_distribution We will only be concerned with measurements that are approximately normal. Nothing will be perfectly normal. In any naturally occurring phenomenon, there will be variation. There are a number of ways to measure the degree to which data varies or is spread out: In addition to the measures of spread discussed earlier, there is variance, and more importantly, STANDARD DEVIATION. Standard deviation is found using a long, tedious process. It is a measure of, on average, the difference between each data item and the mean. It is called, which is the Greek letter for a lower-case s.
M&M Statistics Each member of your group should have a snack size packet of M&Ms. Count and record how many M&M s you have in your snack bag. Ask around. Who had the least? Do you think that that person got ripped off? Did the M&M production company make a mistake in packaging? Explain why or why not. CLASS DATA # of M&Ms Frequency Record the mean, median and standard deviation of the class data. Mean Median Standard Deviation ( ): Calculate the values of: 4. In a NORMAL DISTRIBUTION, A data point is defined to be rare or weird if its value is MORE THAN 2 standard deviation measurements away from the mean in either direction. Did anyone in your group get ripped off? This is different from an outlier that has its own definition.
Algebra2/Trig: How to Draw a Normal Distribution Curve The Normal Distribution 1. Use the calculator to determine the mean,, and the population standard deviation or. Sometimes the regents will ask for the sample standard deviation, which is called or 2. Draw a horizontal line and a bell curve above it, getting very close to the axis at the left and right values and the highest at the mean. Draw 3 SD above the mean and 3SD below the mean, leaving spaces for ½ SD in between if necessary. 3. Use the picture to answer the question. Other things the regents people have been known to ask: : standard deviation of the sample *** especially this one ( ) : the sample variance (literally, just square the above value) ( ) : the population variance (literally, just square ) Standard Deviation Problems 1. 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?
2. Mr. Koziol has 17 students in his high school golf club. Each student played one round of golf. The summarized scores of the students are listed in the accompanying table. Determine the mean and population standard deviation. 3. Christina participated in 20 basketball games this season. The scorekeeper recorded the number of shots she attempted in each game. The table to the right shows the number of shots she attempted in the number of games she played. a. Find the mean number of shots that Christina attempted. Find the standard deviation of the shots attempted to the nearest tenth. What is the total number of games in which the number of shots attempted fell outside one standard deviation of the mean? b. Find the population standard deviation of this set of students' scores, to the nearest tenth. How many of the individual students' golf scores fall within one population standard deviation of the mean? 4. 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 5. 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% 6. The weights of the boxes of animal crackers coming off an assembly line differ slightly and form a normal distribution whose mean is 9.8 ounces and whose standard deviation is 0.6 ounce. Determine the number of boxes of animal crackers in a shipment of 5,000 boxes that are expected to weigh more than 11 ounces.
7. A set of normally distributed student test scores has a mean of 80 and a standard deviation of 4. Determine the probability that a randomly selected score will be between 74 and 82. 8. 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 9. A set of scores with a normal distribution has a mean of 32 and a standard deviation of 3.7. Which score could be expected to occur the least often? 1) 26 2) 29 3) 36 4) 40 10. The mean score on a normally distributed exam is 42 with a standard deviation of 12.1. Which score would be expected to occur less than 5% of the time? 1) 25 2) 32 3) 60 4) 67 11. The amount of ketchup dispensed from a machine at Hamburger Palace is normally distributed with a mean of 0.9 ounce and a standard deviation of 0.1 ounce. If the machine is used 500 times, approximately how many times will it be expected to dispense 1 or more ounces of ketchup? 1) 5 2) 16 3) 80 4) 100 12. The heights of the girls in the eleventh grade are normally distributed with a mean of 66 inches and a standard deviation of 2.5 inches. In which interval do approximately 95% of the heights fall? (1) 61 66 inches (3) 63.5 68.5 inches (2) 61 71 inches (4) 66 71 inches 13. A test was given to 120 students, and the scores approximated a normal distribution. If the mean score was 72 with a standard deviation of 7, approximately what percent of the scores were 65 or higher? (1) 50% (3) 76% (2) 68% (4) 84%
14. On a mathematics quiz with a normal distribution, the mean is 8. If the standard deviation is 0.5, what is the best approximation of the percentage of grades that lie between 7 and 9? (1) 5% (3) 68% (2) 34% (4) 95% 15. The heights of a group of girls are normally distributed with a mean of 66 inches. If 95% of the heights of these girls are between 63 and 69 inches, what is the standard deviation for this group? (1) 1 (3) 3 (2) 1.5 (4) 6 16. The students scores on a standardized test with a normal distribution have a mean of 500 and a standard deviation of 40. What percent of the students scored between 420 and 580? (1) 47.5% (3) 95% (2) 68% (4) 99.5% 17. On a standardized test, the mean is 83 and the standard deviation is 3.5. What is the best approximation of the percentage of scores that fall in the range 76 90? (1) 34 (3) 95 (2) 68 (4) 99 18. On a standardized test with a normal distribution, the mean is 88. If the standard deviation is 4, the percentage of grades that would be expected to lie between 80 and 96 is closest to (1) 5 (3) 68 (2) 34 (4) 95 19. The heights of the members of a high school class are normally distributed. If the mean height is 65 inches and a height of 72 inches represents the 84th percentile, what is the standard deviation for this distribution? (1) 7 (3) 12 (2) 11 (4) 137
Bonus: How Standard Deviation is actually calculated. How to calculate standard deviation (non-grouped data) Find the mean of the data. This is called x-bar or. List each data item. The 1 st datum is called, the 2 nd datum is called, etc and all of them collectively are called.) Find the difference between each data item and the mean. Your answer is allowed to be negative here, but it doesn t actually matter. Square the difference you got in the last step. ( ) Add up all of the squares. ( ). The is the Greek letter for a Capital S, and is used in mathematics to represent the word sum. Divide your last answer by the number (n) of data items. Take the square root of the answer you just got. ( ) ( ) Example 1: 80, 82, 85, 90, 100 ( ) 80 82 85 90 100 ( ) ( ) ( ) Example 2: 7, 12, 15, 20, 21, 26 ( ) 7 12 15 20 21 26 ( ) ( ) ( )
How to calculate standard deviation (grouped data, where frequencies are given) Find the mean of the data. This is called x-bar or. Make sure when you do this, you make sure to take the frequencies into account. List each data item. (still called.) List each data item s frequency (called.) Find the difference between each data item and the mean. (Still called ) Multiply each answer in the previous step by the frequency of each data item. ( ( )) Square the difference you got in the last step. ( ( )) Add up all of the squares. ( ( )). Divide your last answer by the number (n) of data items. Take the square root of the answer you just got. ( ( )) ( ( )) Example 3 80 2 ( ) ( ( )) Example 4 80 2 ( ) ( ( )) 85 3 85 3 87 1 87 1 88 6 88 6 100 1 100 1 ( ( )) : ( ( )) : ( ( )) ( ( )) ( ( )) ( ( ))
Algebra2/Trig: Chapter 15-6 15-9 Bivariate Data and Regressions Univariate data is a statistic that has 1 number to represent the data. Bivariate data is a statistic is that has 2 numbers that represent the data. Bivariate data can be graphed using a scatterplot. A scatterplot graphs pairs of points. Then, a BEST-FIT equation can be found that follows the trend of where the data is going. This best-bit equation can be a line, an exponential curve whatever. Finding a Best-Fit Line (REFERENCE) What you re doing What you should see Press. In L 1, put the values in the 1 st column of the table. In L 2, put the values in the 2 nd column. Quit. Press Turn on the stat plot. Press. Make it look like the display shown. Press. Then press the key until you are on the thing that says DiagnosticOn. Press. Press. Choose one of the best-fit equations (numbered 4, 5, 6, 7, 9, 0, A). The regents SHOULD tell you which to use!! Then press. You should get a screen that looks SOMETHING like this: Type your equation and press. Don t press.
Most of the time, the Regents people will tell you what kind of curve models the data linear, exponential, power, logarithmic. But if they don t, there is a mathematical value called the correlation coefficient that indicates how well the chosen curve fits the given data. The correlation coefficient is called r. The sign of r indicates the slope. Positive slope means positive r. The closer r is to, the better the two variables are correlated. We say the fit is better. You will usually be given a set of data, asked to determine a specific regression equation, and then be asked to make a prediction about the data using the equation that comes up. BE CAREFUL OF YOUR ROUNDING. The regents will deduct a point every time your round incorrectly! Example: 1. In a mathematics class of ten students, the teacher wanted to determine how a homework grade influenced a student s performance on the subsequent test. The homework grade and subsequent test grade for each student are given in the accompanying table. a. Give the equation of the linear regression line for this set of data. b. A new student comes to the class and earns a homework grade of 78. Based on the equation in part a, what grade would the teacher predict the student would receive on the subsequent test, to the nearest integer? 2. Water is draining from a tank maintained by the Yorkville Fire Department. Students measured the depth of the water in 15- second intervals and recorded the results in the accompanying table. a. Write the power regression equation for this set of data, rounding all values to the nearest ten thousandth. b. Using this equation, predict the depth of the water at 2 minutes, to the nearest tenth of a foot.
3. The accompanying table shows wind speed and the corresponding wind chill factor when the air temperature is 10ºF. a. Write the logarithmic regression equation for this set of data, rounding coefficients to the nearest ten thousandth. b. Using this equation, find the wind chill factor, to the nearest degree, when the wind speed is 50 miles per hour. Based on your equation, if the wind chill factor is 0, what is the wind speed, to the nearest mile per hour? 4. In a Jean invested $380 in stocks. Over the next 5 years, the value of her investment grew, as shown in the accompanying table. Write the exponential regression equation for this set of data, rounding all values to two decimal places. Using this equation, find the value of her stock, to the nearest dollar, 10 years after her initial purchase. 5. In the physics lab, Thema determined the kinetic energy, KE, of an object at various velocities, V, and found the linear correlation coefficient between KE and V to be +0.8. Which graph shows this relationship? 1) 2) 6. A linear regression equation of best fit between a student s attendance and the degree of success in school is. The correlation coefficient, r, for these data would be 1) 2) 3) 4) 3) 4)
6. 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. 7. The accompanying table shows the number of bacteria present in a certain culture over a 5-hour period, where x is the time, in hours, and y is the number of bacteria. 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. Write an exponential regression equation for this set of data, rounding all values to four decimal places. Using this equation, determine the number of whole bacteria present when x equals 6.5 hours. 8. Kathy swims laps at the local fitness club. As she times her laps, she finds that each succeeding lap takes a little longer as she gets tired. If the first lap takes her 33 seconds, the second lap takes 38 seconds, the third takes 42 seconds, the fifth takes 50 seconds, and the seventh lap takes 54 seconds, state the power regression equation for this set of data, rounding all coefficients to the nearest hundredth. Using your written regression equation, estimate the number of seconds that it would take Kathy to complete her tenth lap, to the nearest tenth of a second.
Algebra2/Trig: End of Chapter Clean Up and Review Do Now: The breaking strength, y, in tons, of steel cable with diameter d, in inches, is given in the table below. On the accompanying grid, make a scatter plot of these data. Write the exponential regression equation, expressing the regression coefficients to the nearest tenth. a) Using your equation, determine the breaking strength of a 3 inch cable, rounded to the nearest ton. b) Using your equation, determine the diameter of a cable (to the nearest hundredth of an inch) that has a breaking strength of 300 tons. 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. a. Write an exponential regression equation, rounding the calculated values to the nearest ten-thousandth. b. Write a logarithmic regression equation, rounding the calculated values to the nearest ten-thousandth. c. Which is better, a or b? Explain why. d. Algebraically using the better regression equation, determine during which trial would take to return 8 coins.
Percentiles A percentile is a score which is above a certain percentage of all of the values in the data set. For example, the 93rd percentile (% ile) is the value under 93% of the data lies. 1. The heights of the members of a high school class are normally distributed. If the mean height is 65 inches and a height of 72 inches represents the 84th percentile, what is the standard deviation for this distribution? 1) 7 2) 11 3) 12 4) 137 2. 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 More problems 3. 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 4. On a standardized test, Cathy had a score of 74, which was exactly 1 standard deviation below the mean. If the standard deviation for the test is 6, what is the mean score for the test? 1) 68 2) 71 3) 77 4) 80 5. In a normal distribution, and when represents the mean and represents the standard deviation. The standard deviation is 1) 10 2) 20 3) 30 4) 60
The students scores on a standardized test with a normal distribution have a mean of 500 and a standard deviation of 40. What percent of the students scored between 420 and 580? 1) 47.5% 2) 68% 3) 95% 4) 99.5%