MATH NATION SECTION 9 H.M.H. RESOURCES

Transcription

1 MATH NATION SECTION 9 H.M.H. RESOURCES

2 SPECIAL NOTE: These resources were assembled to assist in student readiness for their upcoming Algebra 1 EOC. Although these resources have been compiled for your convenience from the recently adopted textbook materials from Houghton Mifflin Harcourt, digital versions of these materials can also be accessed via the textbook link found in the employee portal. Please be reminded that these materials are copyrighted and should not be posted on school or private websites without prior written permission from the publisher.

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4 9-1 Measures of Center and Spread Reteach You can represent many values in a data set with just one central number. That central number may be the mean or the median. Find the mean by adding the values and dividing by how many values are in the set. Find the median by arranging the values in order and finding the middle value. Example For the data set 6, 10, 8, 13, 0, 9, 5 Find the mean: and the mean. Find the median: 5, 6, 8, 9, 10, 13, 0. The middle number is 9 the median. If a data set has two middle numbers, the median is the average of those two numbers or the number that is halfway between them. With a graphing calculator you can find several statistics about a data set. Example Find statistics about this data set: 13, 5, 9, 11, 3, 8, 7,, 18, 3. Step 1: Use STAT and EDIT to enter the values into L 1. Check your entries for accuracy. Step : Use STAT and CALC to see the 1-Var Stats by pressing ENTER twice. x 13.9 x is the symbol for the mean. The mean is X 139 X is the symbol for the sum of the values. The sum is 139. Skip down two to x 7.5 (rounded). x (say sigma x ) is the standard deviation. n 10. n is the number of values. You entered 10 values. Min X tells you that the minimum, or lowest value, in the set is. Q1 8 tells you that the first quartile is 8. Quartiles divide the set into 4 quarters. Med 1 tells you that the median, or second quartile, is 1. Q3 3 tells you that the third quartile is 3. max X 5 tells you that the maximum, or highest value, in the set is 5. To find the range, find maximum minimum. The range in this set is 5 3. To find the interquartile range, find Q3 Q1. The interquartile range is Range and standard deviation are measures of the spread of the data set. Find each statistic for this data set: 5, 1,, 15, 17, 13, 5, 34, 7, mean. median 3. range 4. first quartile 5. interquartile range 6. standard deviation 69

5 9-1 Measures of Center and Spread Practice and Problem Solving: Modified Two students, Brad and Jin, had the test scores shown below. Use their data for The first one is done for you. Brad: 70, 76, 78, 80, 90, 94, 94, 98 Jin: 80, 8, 84, 84, 86, 86, 88, Find Brad s mean test score.. Find Jin s mean test score. 85 _ 3. Find Brad s median test score. 4. Find Jin s median test score. _ 5. Find Brad s range. 6. Find Jin s range. _ 7. Find Brad s first and third quartiles. 8. Find Jin s first and third quartiles. _ 9. Find Brad s interquartile range. 10. Find Jin s interquartile range. _ Use your statistics from 1 10 to solve. The first one is done for you. 11. In what ways are Brad s and Jin s test scores similar? Possible answer: Their means are equal and their medians are equal. 1. In what ways are Brad s and Jin s test scores different? 13. Which of the two students would you consider a more consistent test taker? Explain your thinking. 14. One of the students has test scores with a standard deviation of 3 and the other has test scores with a standard deviation of 9.6. Without calculating, how can you tell which student has each standard deviation? 70

6 9- Data Distributions and Outliers Reteach To make a dot plot from a data set, put an X above the number line for each value. Skewed to the right Symmetric Skewed to the left Find the mean of a data set; add the numbers and divide by the number of items in the set. Find the median: list the items in order and find the middle number (or the average of the two middle numbers). Find the range: calculate maximum minimum. Find the IQR (the interquartile range): calculate Q 3 Q 1 (quartile 3 minus quartile 1). A data set may have an extreme value that does not fit in with the other values in the set. This outlier is sometimes removed from the data set because it distorts the central value. A value, x, is an outlier if it is below ( Q 3 1 IQR ) or above ( Q 3 3 IQR). Example Find the outliers in this data set:, 11, 14, 18, 1, 43. Q 1 11, Q 3 1, IQR 10 1) Find Q 1, Q 3, and the IQR using a calculator. ( Q 3 1 IQR) ) Calculate ( Q 3 1 IQR) and ( Q 3 3 IQR). ( Q 3 3 IQR) ) An outlier falls outside 4 to is an outlier in this data set. Use this data set to answer the questions below: 0, 5, 35, 40, 45, 50, 50, 55, 55, 55, 60, 60, 60, Make a dot plot of the data. Identify the plot as symmetric, skewed left, or skewed right.. Find the mean. 3. Find the median. 4. Find the IQR. 5. If the number 100 were added to the data, would it be an outlier? Explain. 71

7 9- Data Distributions and Outliers Practice and Problem Solving: Modified Identify each dot plot as symmetric, skewed to the left, or skewed to the right. The first one is done for you. 1.. skewed to the left _ _ The table below shows the scores of ten golfers in a tournament. Use the data for Problems The first one is done for you Find the mean and median. 6. Find the range and interquartile range. mean: 7.8; median: 74 _ 7. Make a dot plot for the data. 8. Identify the dot plot as symmetric, skewed to the left, or skewed to the right. 9. Suppose an 11th golfer with a score of 95 is added to the tournament scores. Which of the statistics from Problems 5 and 6 would change? 10. If a score of 95 were added, would it be an outlier? Explain. 7

8 9-3 Histograms and Box Plots Reteach Make a Histogram The estimated miles per gallon for selected cars are given: {6, 8, 3, 33, 6, 15, 1, 35, 17, 18, 5, 9, 30, 6, 7, 30, 4, 5, 4, 3, 5, 19,, 3, 5, 31, 8, 3, 7, 3, 4, 0, 38, 44, 18}. Step 1: Find the difference between the greatest and least values. Try different widths for your intervals to determine the number of bars in the histogram. Use the difference to decide on intervals. Car Gas Mileage mi/gal Frequency Step : Use the difference to decide on intervals. Try different widths for your intervals to determine the number of bars in the histogram. Step 3: Create the frequency table. Step 4: Use the frequency table to create the histogram. Draw each bar to the corresponding frequency. 1. Use the data set above to make a frequency table with intervals. Then make a histogram. Box Plots Consider the data set {3, 5, 6, 8, 8, 10, 11, 13, 14, 19, 0}. Step 1: To make a box plot, first identify the five key numbers. Step : Plot the five numbers above a number line. Draw a box so that the sides go through Q1 and Q3. Draw a line through the median. Connect the box to the minimum and maximum.. Write the following data in order: 9, 11, 18, 1, 18, 14, 5 3. Minimum:, Q1:, Median:, Q3:, Maximum: 4. Using the data set from Problem, draw the box plot. 73

9 9-3 Histograms and Box Plots Practice and Problem Solving: Modified The heights of players at a basketball game are given in the table below. Use the data for 1 7. Players Heights (in.) Use the data to make a frequency table.. Use your frequency table to make The first one is started for you. a histogram for the data. Players Heights Heights (in.) Frequency Which interval has the most players? 4. Describe the shape of the distribution. _ 5. Use the midpoints of each interval to estimate the mean height of a player. Use the box plot for 6 9. The first one is done for you. 6. Find the median age. 7. Find the range. 8 _ 8. Find the interquartile range. 9. Find the age of the oldest player. _ 74

10 9-4 Normal Distributions Reteach You can take a table of relative frequencies showing measurement data and plot the frequencies as a histogram. When the intervals for the histogram are very small, the result is a special curve called a normal curve, or bell curve. Data that fit this curve are called normally distributed. When you know the median (the y-height at 0) and the standard deviation (marked as 1,, 3) of the data, you can use the curve to draw conclusions and make predictions about the data. Here are some statements that are true about this special curve. The mean and the median are the same the center of the curve at its highest point. The curve is symmetric. If you draw a vertical line at the median, the two sides match. About 68% of all the data is within 1 standard deviation ( 1 to 1) from the mean. About 95% of all the data is within standard deviations ( to ) from the mean. About 99.7% of all the data is within the 3 standard deviations ( 3 to 3) from the mean. Example The scores for all the Algebra 1 students at Miller High on a test are normally distributed with a mean of 8 and a standard deviation of 7. What score is 1 standard deviation above the mean? What score is 1 standard deviation below the mean? What percent of students made scores between 75 and 89? 68% What percent of students made scores above 89? 13.5%.5% 16% What is the probability that a student made a score above 96? 8 (7) 96 This score is more than standard deviations from the mean. The probability is.5%. The scores for all the sixth graders at Roberts School on a statewide test are normally distributed with a mean of 76 and a standard deviation of What score is 1 standard deviation. What score is standard deviations above the mean? below the mean? _ 3. What percent of the scores were 4. What percent of the scores were below 56? above 86? _ 5. What is the probability that a student made a score between 66 and 86? 75

11 9-4 Normal Distributions Practice and Problem Solving: Modified Solve. The first one is started for you. 1. The graph below shows a normal curve that has been divided into eight sections. Each section represents one standard deviation above or below the mean. Fill in the percentage of the area under the curve that each section contains.. Find the sum of the percents written above the curve in Problem 1. Explain why that sum makes sense. 3. Use your curve above to complete each sentence with the correct percent for normally distributed data. (a) % lie within 1 standard deviation of the mean. (b) % lie within standard deviations of the mean. (c) % lie within 3 standard deviations of the mean. A company selling light bulbs claims in its advertisements that its light bulbs average life is 1000 hours. In fact, the life span of these light bulbs is normally distributed with a mean of 1000 hours and a standard deviation of 100 hours. Use this information for Problems 4 7. The first one is done for you. 4. Find the probability that a randomly chosen light bulb will last between 1000 and 1100 hours. 34% 5. Find the probability that a randomly chosen light bulb will last less than 900 hours. 6. Find the probability that a randomly chosen light bulb will last more than 100 hours. 7. Find the probability that a randomly chosen light bulb will last between 800 and 1100 hours. 76