Exam (elaborations) Milestone Unit 4 Statistics Sophia

Exam (elaborations) Milestone Unit 4 Statistics Sophia 1 The scores of the quizzes of five students in both English and Science are: English Science Student 1 6 8 Student 2 5 5 Student 3 9 6 Student 4 4 7 Student 5 8 9 For English, the mean is 6.4 and the standard deviation is 2.0. For Science, the mean is 7 and the standard deviation is 1.6. Using the formula below or Excel, find the correlation coefficient, r, for this set of scores. Answer choices are rounded to the nearest hundredth. 0.50 0.23 0.05 0.42 RATIONALE In order to get the correlation, we can use the formula: Correlation can be quickly calculated by using Excel. Enter the values and use the function "=CORREL(". Milestone Unit 4 Statistics Sophia CONCEPT Correlation 2 This scatterplot shows the number of hours a student slept every night and his or her grade point average. The equation for the least-squares regression line to this data is: ŷ = 0.375x + 1.33. What is the predicted GPA for a student who sleeps 2.5 hours per day? Answer choices are rounded to the hundredths place. 2.64 2.46 2.08 2.27 RATIONALE In order to get the predicted GPA when the hours of sleep is equal to 2.5, we simply substitute the value 2.5 in our equation for x. So we can note that: CONCEPT Predictions from Best-Fit Lines 3 Data for price and thickness of soap is entered into a statistics software package and results in a regression equation of ŷ = 0.4 + 0.2x. What is the correct interpretation of the slope if the price is the response variable and the thickness is an explanatory variable? The price of the soap decreases by $0.20, on average, when the thickness increases by 1 cm. The price of the soap increases by $0.40, on average, when the thickness increases by 1 cm. The price of the soap decreases by $0.40, on average, when the thickness increases by 1 cm. The price of the soap increases by $0.20, on average, when the thickness increases by 1 cm. RATIONALE When interpreting the linear slope, we generally substitute in a value of 1. So we can note that, in general, as x increases by 1 unit the slope tells us how the outcome changes. So for this equation we can note as x (thickness) increases by 1 cm, the outcome (price) will increase by $0.20 on average. CONCEPT Interpreting Intercept and Slope 4 Which of the following scatterplots shows an outlier in both the x- and y-direction? RATIONALE To have an outlier in the x-direction and y-direction the outlier must be outside of the range of y data and outside the range of x-data. This outlier is below in the y-direction and to the left in the x-direction. CONCEPT Outliers and Influential Points 5 A correlation coefficient between average temperature and ice cream sales is most likely to be __________. between 1 and 2 between 0 and –1 between –1 and –2 between 0 and 1 RATIONALE In general as temperature increases, tastes for ice cream goes up. So the correlation should be positive, which would be between 0 and 1. CONCEPT Positive and Negative Correlations 6 Which statement accurately describes the data's form, direction, and strength from the scatterplot below? Form: Linear Direction: Positive Strength: Weak Form: Linear Direction: Negative Strength: Weak Form: Linear Direction: Positive Strength: Moderate Form: Linear Direction: Negative Strength: Moderate RATIONALE If we look at the data, it looks as if a straight line captures the relationship, so the form is linear. The slope of the line is positive, so it is increasing. Finally, even though the direction is clear, the data points are less clustered in a line or curve, so the strength is moderate. CONCEPT Describing Scatterplots 7 A basketball player recorded the number of baskets he could make depending on how far away he stood from the basketball net. The distance from the net (in feet) is plotted against the number of baskets made as shown below. Using the best-fit line, approximately how many baskets can the player make if he is standing ten feet from the net? 5 baskets 9 baskets 3 baskets 8 baskets RATIONALE To get a rough estimate of the number of baskets made when standing 10 feet from the net, we go to the value of 10 on the horizontal axis and then see where it falls on the best-fit line. This looks to be about 5 baskets. CONCEPT Best-Fit Line and Regression Line 8 For this scatterplot, the r 2 value was calculated to be 0.9382. Which of the following set of statements is true? About 94% of the variation in beach visitors can be explained by a positive linear relationship with daily temperature. The correlation coefficient, r, is 0.969. There is no strong correlation in the linear association between beach visitors and daily temperatures. The correlation coefficient, r, is 0.880 About 94% of the variation in daily temperature can be explained by a positive linear relationship with beach visitors. The correlation coefficient, r, is 0.880 About 94% of the variation in beach visitors is explained by a negative linear relationship with daily temperatures. The correlation coefficient, r, is 0.969. RATIONALE The coefficient of determination measures the percent of variation in the outcome, y, explained by the regression. So a value of 0.9382 tells us the regression with temperature, x, can explain about 94% of the variation in visitors, y. We can also note that r = . CONCEPT Coefficient of Determination/r^2 9 A bank manager declares, with help of a scatterplot, that the number of health insurances sold may have some association with the number of inches it snows. How many policies were sold when it snowed 2 to 4 inches? 210 240 350 470 RATIONALE In order to find the total number of policies between 2 and 4 inches, we must add the three values of 10 in that interval. At 2 inches, there were 100 policies. At 3 inches, there were 110 policies. At 4 inches, there were 140 policies. So the total is 100 + 110 + 140 = 350 policies. CONCEPT Scatterplot 10 For ten students, a teacher records the following scores of two assessments, Quiz 1 and Test. Quiz 1 (x) Test (y) 15 20 12 15 10 12 14 18 10 10 8 13 6 12 15 10 16 18 13 15 Mean 11.9 14.3 Standard Deviation 3.3 3.5 The correlation of Quiz 1 and Test is 0.568. Given the information below, what is the slope and y-intercept for the least-squares line of the Quiz 1 scores and Test scores? Answer choices are rounded to the hundredths place. Slope = 0.60 y-intercept = 7.16 Slope = 0.60 y-intercept = 1.22 Slope = 0.54 y-intercept = 1.71 Slope = 0.54 y-intercept = 1.22 RATIONALE We first want to get the slope. We can use the formula: To then get the intercept, we can solve for the y-intercept by using the following formula: y with hat on top equals b subscript 0 plus b subscript 1 x We know the slope, b subscript 1, and we can use the mean of x and the mean of y for the variables x and y with hat on top to solve for the y-intercept, b subscript 0. CONCEPT Finding the Least-Squares Line 11 Jesse takes two data points from the weight and feed cost data set to calculate a slope, or average rate of change. A hamster weighs half a pound and costs $2 per week to feed, while a Labrador Retriever weighs 62.5 pounds and costs $10 per week to feed. Using weight as the explanatory variable, what is the slope of a line between these two points? Answer choices are rounded to the nearest hundredth. $0.13 / lb. $7.75 / lb. $4.00 / lb. $6.25 / lb. RATIONALE In order to get slope, we can use the formula: s l o p e equals fraction numerator y 2 minus y 1 over denominator x 2 minus x 1 end fraction. Using the information provided, the two points are: (0.5 lb., $2) and (62.5 l