TAKS Tutorial Objectives 5 & 9 part 1

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Objective 9 on Probability & Statistics is tested at all levels of TAKS Today, we will work with:  Central measures of tendency (mean, median, mode, range)  Compound probability

Let’s start with MEDIAN Median is the value that is in the middle of a data set. Let’s see a visual depiction of median.

Place the buildings in order by height. Now, eliminate a smallest and a largest building until you get to the middle. MEDIAN --MIDDLE XXX X XX

On the TAKS test, you will be given a set of numbers in a data set. The first thing you want to do is to put those numbers in numerical order from smallest to largest. Then, just like we did with the buildings, you want to cross out one smallest and one largest number IN PAIRS until you get to the middle. What happens if you have nothing left after eliminating the pairs? Use the last pair, the two middle values, and find their average. You want the number in the middle of them.

Let’s move on to MODE Mode is the value that occurs most often in a data set. Let’s see a visual depiction of mode.

Tally how many times each height occurs. All of the buildings are tallied. The height that occurred most often is 180 meters.

There are some facts you need to remember about Mode.  The value must occur more than once to be a mode  If all values occur only once, there is no mode.  If more than one value occurs most often, each are modes.

RANGE is the difference between the largest and the smallest data value. With your data in order from smallest to largest, just subtract. Largest value – smallest value = Range

The MEAN (or AVERAGE) value is where we “play fair” and act like Robin Hood. We take from the largest values and give to the smallest values so they all have an equal amounts. This process is what happens to your grades—we take points from your highest grades to give to your lowest grades until we get equality. That is why zeroes hurt your grade so much!

We need to look for a value where each of these buildings would be equal in height. How about right here? This extra Could go here

We need to look for a value where each of these buildings would be equal in height. How about right here? This extra Could go here

We need to look for a value where each of these buildings would be equal in height. How about right here? The extra Could go here And here

We need to look for a value where each of these buildings would be equal in height. Right here looks just fine! -30m 150 m +30m 150 m -20m 150 m +20m 150 m -30m 150 m -10m 150 m +40m 150 m

It is much easier to find the MEAN by just adding all the data values together and then dividing by the number of values. Then, you don’t have to figure out how to divide everything up evenly. The process does it automatically.

This problem was on the April 2006 test. You can see that having all 4 measures as options requires you to know them all and perform them all.

Task 1:Put the car prices in order from least to greatest. Fortunately for us, they already are in that order. The number there most often is $995 so that is the Mode. $995 Range: 1495 – 600 = 895 $895 Lightly cross out the smallest & largest values in pairs to find the median. $750 So now, bring out the calculator and find the mean. $863

In Middle School, you learned to find simple probability. You then extended that knowledge to compound probability. Compound probability is where you find the likelihood of one event happening and multiply it by the likelihood of the next event happening.

For instance, you just dumped all of you clean socks in your sock drawer without folding them. One morning before school, you get dressed in the dark. What is the probability that when you pull one sock out of the drawer, put it on your foot, and then pull out a second sock and put it on your other foot, that you have the same color socks on your feet? Before we can answer this question, we need to know how many socks we have of each color.

 If we only have one color of socks— say they are all black—the probability of having the same color socks on our feet is 1. We definitely have a matched pair!  If every sock in the drawer is a different color, then the probability of a matched pair is 0.  All other probabilities are some fractional value between 0 and 1.

Let’s say we have 8 black socks, 4 brown socks, 6 white socks, and 4 blue socks in our drawer. What is the probability that we have a pair of black socks on our feet? It would mean that we would have had 8 chances out of 22 total socks to get a black sock the first time.

Remember, that sock goes on our foot, so now we have 21 socks in the drawer. I have 1 of the black socks on my foot, so there are 7 black socks left out of the 21 socks still in the drawer. To find the probability that I have the pair of black socks on my feet, I need to multiply the two single probabilities together.

Since most of you don’t like fractions anyway, let your calculator do the multiplying for you. Just remember to put parentheses around your fractions! We do NOT want that decimal. So, use the calculator again to convert the probability into a fraction. Remember, MATH Frac We have only a 4 in 33 chance of having a matched pair of socks. Not very good odds!

This question was on the 2004 test. Although Jamal’s name is 5 letters long, there are only 4 different letters used. He has a 4 out of 26 probability of drawing a letter from his name.

This question was on the 2004 test. Half of the number tiles are odd and half of them are even. There are 5 out of 10 odd numbers: 1, 3, 5, 7. and 9.

This question was on the 2004 test. Multiply the probability of each event happening.

Today, we are also going to take a look at Objective 5 Non-linear Functions Since this test covers Algebra I material, we are talking about Quadratic Functions and a small amount on Exponential Functions. That doesn’t mean that you won’t see other functions that you have studied mentioned in the test questions.

With respect to the quadratic functions: Today, we will work with:  Parameter changes on the parent function, y=x 2 (which is part of Objective 2)  Comparing the graphs of any 2 quadratic functions  Solving quadratic equations  Roots, solutions, zeroes, x-intercepts

In Objective 2, the expectation is that you know the equation of the linear and quadratic parent functions…

In Objective 5, the expectation is that you will know the effect that changes on the parent function will have on its graph.  y = x 2 y = ax 2 + c  Translation: up when c is positive  Translation: down when c is negative  Reflection: when a is negative  Dilation: wider when a is smaller than 1  Dilation: narrower when a is larger than 1

2006 test You may only translate up or down. Look at the graph you have. Find where x = 6. Now, find the point (6, 7)

To get from the point on the given graph (6, 11) to the point (6, 7), you need to translate down 4 units. If the entire graph moves down 4 units, the new y- intercept/vertex is at (0, -5) 4 4

Now, look at the answer choices. Which function has the correct y-intercept? (Also, note that two of the functions open downward because the 1/3 is negative. Our graph opens upward.) To check your answer, enter the function in the calculator and consult the table to make sure (6, 7) is there.

2006 test Analyze the first function. The coefficient is negative—so the graph opens downward. ¾ is a number smaller than 1 so that parabola is wider than the parent. 4/3 is a number larger than 1 so that parabola is narrower than the parent. Consequently, the ¾ makes the first parabola wider than 4/3.

A big MUST to remember… These expressions/words all mean the same thing!!!!  roots  solutions  zeroes  x-intercepts Know this information!!!!!!!

2006 test If you know that roots are x- intercepts, you just need to look at the graph to find where the parabola crosses the x-axis. All x-intercepts have zero for a y-coordinate!! Between 2 and 3 Between 6 and 7

2003 test You have several options for solving this problem! Option 1: solve graphically, finding the x-intercepts Move 36 to the left side of the equation (= 0 “zeroes”) and enter in y 1 on the calculator.

2003 test Adjust the window—you don’t need that much of the x-axis. You can approximate the x-intercepts. About -1/3 About 5/3

2003 test Adjust the window—you don’t need that much of the x-axis. Or you can CALC the zero About -1/3 About 5/ is -1/3 This option is the only one with -1/3 If you don’t like these fractions, you can change them all to decimals. Just to show you the other solution… … is 1 2/3 which is 5/3

2003 test Here is another option… Option 2: solve graphically, finding the intersection. Enter the left side of the equation in y 1 and the right side in y 2. You will need to adjust the window so y max is higher than 36. Use the CALC feature and find the intersect(ion) Once again, is -1/3 and C is the only option with that solution.

2003 test And a third option! Solve Algebraically

More Practice for these topics…  Here are some problems for you to do…

11’03 8 An automobile dealer is analyzing a frequency table identifying the number of vehicles of each color sold during the last 6 months. Which measure of data describes the most popular color of vehicle sold? F Mean G Median H Mode J Range

10’04 34 Nicholas earned the following grades on his science exams: 83, 88, 87, and 83. If Nicholas scores a 90 on his last exam, which measure of central tendency will give him the highest score? F Mode G Median H Range J Mean