# Final Review Sheet. Mod 2: Distributions for Quantitative Data

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1 Things to Remember from this Module: Final Review Sheet Mod : Distributions for Quantitative Data How to calculate and write sentences to explain the Mean, Median, Mode, IQR, Range, Standard Deviation, Frequency (n), Q1 and Q3 Shapes of Histograms and Dot Plots Know the difference between categorical and quantitative variables Make a boxplot using data How to calculate whether something is an outlier or not Write an essay describing a data set by interpreting all of the sample statistics and graphs. Practice questions 1. Use the following set of data: 7, 9, 9, 10, 11, 11, 1, 1, 1, 1, 15, a. Find the Median Q : 11.5 b. Find the quartiles Q 1 and Q 3. Now find the Interquartile Range (IQR). Q1=9.5, Q=11.5, Q3=1, IQR=.5. Create a boxplot from the data; don t forget to calculate if there are any outliers!!.5 Boxplot of C C Using the following data set: 1, 1,, 3, 3, 5, 6 a. Find the Mean 3 b. Find the Range 5 c. Find the Standard Deviation 1.9

2 4. What are the four shapes discussed in chapter? For each shape, draw a histogram and boxplot with that shape. Skewed Left (tail on left), Skewed Right (tail on right), Bell Shaped (Symmetric, Normal), Uniform (Rectangular shape) 5. If you are asked to describe a data set in an essay from graphs and sample statistics, what key things should you discuss? Shape, Center (find best Average), Spread (find how spread out typical numbers are and find two #s that typical numbers fall inbetween), Outliers (should they be removed or not and if removed, what does that do to the shape?) Things to Remember from this Module: Mod 3: Linear Regression How to calculate slope, describe slope as a rate of change with units How to find the equation of a regression line from r, standard deviations and means What does correlation tell us? Facts about correlation, strength and direction How to graph lines Word problems with slope/lines Explanatory/response Variables, know how to determine which is which Interpret r, r, S e and residual plots Lurking variables, be able to identify possible lurking variables Come up with the regression line using formulas Do not extrapolate using your linear regression model Be able to check the 6 regression criteria Practice Questions: 6. In 1998, Ann bought a house for \$184,000. In 009, the house is worth \$305,000. Find the average annual rate of change in dollars per year of the value of the house. Round your answer to the nearest cent. Let x represent number of years after a. Write the data provided into two ordered pairs (8,184000) (19, ) b. Find the slope, include the appropriate label, and interpret the meaning of the slope m = 11000, Housing prices are increasing \$11000 per year c. Find the y intercept and interpret the meaning of the y intercept b = In 1990, (year zero) the price of a house was \$96000

3 d. Find the equation of the line representing Ann s house value y=11000x e. Estimate Ann s house value in 004 \$ Give the following scatterplots a. Write positive, negative, or no correlation next to each scatterplot Plot 1: moderate positive correlation ; Plot : No correlation but nonlinear relationship ; Plot 3: No correlation No relationship ; Plot 4: Strong Negative Correlation b. Which graph has the strongest linear correlation Plot 4 c. Which graph has the strongest non-linear correlation Plot d. The four correlation coefficients for the scatterplots shown are , , , and Match the correlations to the plots. Plot 1: ; Plot & 3: or ; Plot 4: Does a strong positive correlation PROVE that one variable causes the other variable to respond? Does no correlation always imply there is no relationship? Give an example of two variables that confirms your answers. A strong Positive correlation indicates that the two variables have a positive linear relationship. That is to say as the explanatory variable increases, the response variable also increases. It does not imply that one variable causes the other because there may be lurking variables involved that may influence the explanatory and response variables. No correlation simply means no linear relationship. There can be a nonlinear relationship like a quadratic, logarithmic or exponential relationship.

4 9. For an essay question on a relationship between two variables, what key things should you talk about? You should talk about the scatterplot, r value, r squared, Standard Deviation of the Residual Errors, the type of relationship (correlation or nonlinear or none) and the strength and direction of the relationship (strong, moderate, weak, positive, negative). If there is correlation, you should also find the regression line of best fit and interpret the slope and y intercept. You should then use the residual plots to check the 6 regression criteria. 10. We looked at 40 randomly selected men to analyze the relationship between the weight of a man and his BMI (Body Mass Index). Minitab found the following graphs and statistics. 34 Scatterplot of BMI vs WT BMI Correlation of WT and BMI = The regression equation is BMI = WT WT i. What does the scatterplot tell us about the relationship between the weight of a man and his Body Mass Index? There is a moderate to strong positive correlation between weight and BMI. ii. A trainer said that if a man is heavy, it will cause him to have a large BMI. Does the data support this statement? It does not. Heavy men do tend to have larger BMI s, but that does not mean that one causes the other. We cannot make causation statements. iii. What is the r value and what does it tell us? The r value is and this tells us that there is moderate to strong positive correlation between weight and BMI iv. What is the r value and what does it tell us? List other lurking variables that might influence BMI besides weight? R squared is 0.64 or 64%. This means that 64% of the variability in BMI can be accounted for by its linear relationship to the weight. There can be lurking variables such as height, amount of muscle, nutrition, etc.

5 v. Use the regression line to predict the BMI of a man that weighs 0 pounds? How accurate do you think this prediction is? BMI=30.9 This prediction is moderately accurate since the regression line fit the data reasonably well. vi. Can we use the regression line to predict the BMI of a man that is 100 pounds? Why or why not? No. We should not use the regression line to predict the BMI of a 100 pound man since 100 is not in the scope of the data and would therefore be extrapolation and subject to a lot of error. 11. Find the least squares regression line Regression Lines. Give the equations for slope and y intercept of a regression line. r sy y mx b Where m and b y m x m= b= s y= x x Match each description of a set of measurements to a scatterplot. 1. x = average outdoor temperature and y = heating costs for a residence for 10 winter days Plot x = height (inches) and y = shoe size for 10 adults Plot 1

6 14. x = height (inches) and y = score on an intelligence test for 10 teenagers Plot 15. For the following data: a) Make a scatter plot. (x,y) Scatterplot of C5 vs C4 x y C C b) Draw a line of fit for your scatter plot, find the equation of the line two points (Answers May vary) y=1.3x-0.13 Scatterplot of C5 vs C C c) Use your equation to predict y for x = 13 C4 Y = 16.8 but notice this is extrapolation. Will probably have more error in this prediction 16. What does an outlier do to an r-value? Explain in complete sentences and say how will the r-value change, increase or decrease? An outlier can have a dramatic effect on the shape of a scatterplot. Adding an outlier can sometimes decrease the correlation dramatically. It usually result in decrease in the strength of the linear (or nonlinear relationship). Identify the explanatory and response variables in the following: 17. How the price of a package of meat is related to the weight of the package?

7 a. Explanatory: Weight b. Response: Price 18. Is the distance you walk related to the calories you burn? a. Explanatory: Distance b. Response: Calories 19. Use the following Scatterplot a) The line we will use to describe this scatterplot is: D = -3A + 576, use the linear model to interpret the slope. The slope means that as a person gets one year older, the distance to read the sign is decreasing 3 feet. b) Interpret the y intercept. The y intercept of 576 feet would be the distance to read the sign at age zero. Does not make sense in the context of this problem. Notice (0,576) is not in the scope of the data. Mod 4: Non Linear Modeling Things to Remember from this Module: Create exponential, logarithmic, and quadratic curves with technology, give the scope of the x values, r-squared, standard deviation of the residual errors and use them to assess the fit of the curve to the data. Create residual plots and analyze the 6 regression criteria. Use exponential, quadratic, and logarithmic models to make predictions Use R-squared, and the Standard Deviation of the Residual Errors to determine which curve is the best fit. Practice Problems

8 0. Write an essay on the following topic: How can we use Nonlinear Regression curves to describe the relationship between two variables? How can we use Standard Deviation of the Residual Errors and residual plots to assess which nonlinear function makes the best fit? 1. Use the nonlinear data on the website Analyze the Mother s Age verses Baby weight data. Find an exponential function that fits the data using a LN (Y) transformation. What is the R-squared and Standard Deviation of the Residual Errors? Write sentences explaining each. (Remember the Standard Deviation is wrong in Statcrunch.) Give the residual plot vs the x values and the histogram of the residuals and analyze the 6 regression criteria.. Use the nonlinear data on the website Analyze the Mother s Age verses Baby weight data. Find a natural Log (LN) function that fits the data using a LN (x) transformation. What is the R-squared and Standard Deviation of the Residual Errors? Write sentences explaining each. Give the residual plot vs the x values and the histogram of the residuals and analyze the 6 regression criteria. 3. Use the nonlinear data on the website Analyze the Mother s Age verses Baby weight data. Find a quadratic function that fits the data using Statcrunch. What is the R-squared and Standard Deviation of the Residual Errors? Write sentences explaining each. Give the residual plot vs the x values and the histogram of the residuals and analyze the 6 regression criteria. 4. Which curve was the best fit for the Mother s Age vs baby weight data? 5. Use the exponential function to predict the baby weight if the mom is 18 years old. 6. Use the natural log (LN) function to predict the baby weight if the mom is years old. How much error will be in that prediction? 7. Use the Quadratic function to predict the baby weight if the mom is 30 years old. How much error should we expect in that prediction? 8. Go to the nonlinear data again. Find the quadratic function that best fits the month in 009 verses solar energy in kilowatt hours. What is the x and y coordinate of the vertex. Explain the meaning of both the x coordinate and the y coordinate.

9 Things to Remember from this Module: Mod 5: Probability and Two way Tables How to create two way tables from data OR percentages How to use two way tables to find marginal, conditional, and joint probabilities. Use the two way table below to answer the following questions. Prefer Cats Prefer Dogs Do not like cats or dogs Male Female What percent of the sample preferred dogs? 136/ If a person was female, what is the probability that they preferred cats? 56/ What is the probability of a person being both male and preferring dogs? 89/40 3. What percent of the people were either female or did not like cats or dogs? 137/ Find the probability of a person preferring cats, if we are given that the person is male. 14/ Describe the process of making a two way table if you are only given the grand total and marginal and conditional percentages. Use the marginal (regular) percentages to find the totals. Then use the conditional percentages to fill in the table. You will be multiplying the conditional percentages times the totals.

10 Mod 6: Probability Tables, Expected Values, Empirical Rule, Z-scores Things to Remember from this Module: How to create and use a probability table Compute and interpret the expected value of a probability table Use the Empirical rule to find probabilities for normal quantitative data Calculate and interpret z-scores 35. Look at the following probability table given for the number of missed assignments in Math 075. Find the missing probability. What is the probability that a Math 075 student has 3 or more missing assignments? Find the expected value for the table and write a sentence interpreting the meaning of the expected value. # Missing Assignments Probability ?? Probability of missing 3 or more assignments = 0.61 or 61%. E =.84 On average, students miss.84 assignments per semester. A typical student would probably miss or 3 assignments. 36. The mean average Reading Comprehension SAT score in California in 01 was 489 with a standard deviation of 77. Assuming that the Reading Comprehension SAT scores were normally distributed, draw a normal curve and fill out the curve using the principles of the Empirical Rule. Be sure to include the SAT scores as well as the percentages. What percent of students scored more than a 643 on the Math SAT? What are the two SAT score that separates the middle 99.7% of students? P(scoring more than 643) =.5% 99.7% of students score between 58 and 70 on their Reading Comprehension SAT.

11 37. The mean average Reading Comprehension SAT score in California in 01 was 489 with a standard deviation of 77. Amy got a 700 on her Reading Compreshension SAT. Calculate a Z-score for Amy and write a sentence interpreting the meaning of the Z-score. Was Amy s score unusual? Z =.74 Amy scored.74 standard deviations above the mean average on her Reading Comprehension SAT. This was unusually high.

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