One Variable and Two Variable Statistics

Transcription

1 Name: Class: Date: ID: A One Variable and Two Variable Statistics Short Answer 1. When recommending five mutual funds for you to consider, your financial planner mentions that the typical minimum investment is $2000 since the minimum investment amounts for the five funds are $500, $2000, $2500, $25 000, and $2000, respectively. However, when you key these numbers into a calculator, you find that the average minimum investment is, in fact, $6400. What accounts for this discrepancy? 2. A building maintenance company tracked the number of months that fluorescent tubes lasted in the different offices in a building. Calculate the mean, median, and mode for this set of data. 8, 29, 22, 15, 10, 22, 12, 4, The numbers of service calls a heating company made during the first 11 days of October are listed below. Find the mean, median, and mode for this set of data. 6, 28, 28, 11, 30, 21, 17, 25, 28, 28, The numbers of houses a real-estate agency sold each month from January to October are listed below. Calculate the mean, median, and mode for these sales figures. 8, 3, 16, 6, 24, 13, 16, 16, 2, The ages, in years, of a group of friends are listed below. 29, 33, 36, 48, 50, 51, 53, 53 a) Find the mean, median, and mode of the ages. b) Explain what each of these measures tells you about this group of friends. c) What do the relative values of the mean and median tell you about the group? 1

2 Name: ID: A A consumer magazine evaluated 39 models of bathroom scales. The table below lists the prices for these models (rounded to the nearest dollar). Scale Model Price ($) Scale Model Price ($) EconoHealth A10 50 Superskale EconoHealth A12 50 Superskale EconoHealth B10 50 Superskale EconoHealth E10 28 Superskale EconoHealth Digital SvelteChek EconoHealth E SvelteChek 12400D 48 EconoHealth E SvelteChek HealthSkale SvelteChek HealthSkale SvelteChek Fashion 17 HealthSkale SvelteChek Pro 50 HealthSkale 290 Deluxe 79 SvelteChek Xtra 25 HealthSkale Weighbeter HealthSkale Weighbeter 801D 60 HealthSkale Weighbeter HydroXact Weighbeter HydroXact Weighbeter HydroXact Weighbeter Prowt P10A 120 Weighbeter Prowt Value 35 Weighbeter Basic 12 Prowt Value Find the median, first quartile, and third quartile for the prices of these bathroom scales. 7. Find the range and the interquartile range for the prices of these bathroom scales. 8. Identify the likely outlier in this set of data. X Y Find a regression equation with a coefficient of determination greater than 0.98 for the following set of data: X Y Find a regression equation with a coefficient of determination greater than 0.98 for the following set of data: X Y

3 Name: ID: A Problem 11. The usage of a hospital X-ray machine was monitored for 9 days. The data, rounded to the nearest hour, are listed below. 3, 13, 9, 10, 13, 21,12, 23, 13 a) Calculate the mean, median, and mode for these data. b) Suggest a reason why you might want to exclude the lowest value from the calculations in part a). 12. The following table lists the approximate numbers of residents in 21 Canadian cities in City Population City Population Calgary Saskatoon Edmonton Sault Sainte Marie Halifax St. John's Hamilton Sudbury Kingston Thunder Bay Kitchener/Waterloo Toronto Lethbridge Vancouver London Victoria Ottawa Windsor Regina Winnipeg Saint John a) Find the median, first quartile, and third quartile for these data. b) Determine the range and interquartile range. c) Calculate the mean, standard deviation, and variance. d) What is the z-score for the population of Windsor? e) What is the z-score for the population of Toronto? 13. A study was done with a group of university students to determine if there was a correlation between the amounts of sleep they got and their academic performance. The following table lists some data from the study. Student A B C D E F G H I J K L Hours of Sleep Average Mark a) Make a scatter plot of these data. b) Using a graphing calculator or a spreadsheet program, determine the correlation coefficient. c) What would you conclude about the relationship between the two variables? 3

4 Name: ID: A 14. The coach of the Statsville football team wants to determine if there is a relationship between how fast players can run 60 m and how far they can throw the football. The results for the Statsville players were as follows. Player Sprint Time (s) Throwing Distance (m) Jon H Tom M Sarjay P Brandon F Tyler C Steve K Matt H Robin L Alex H Mike N Ankit K Scott R a) Using technology, create a scatter plot of sprint times versus throwing distances. b) Perform a linear-regression analysis of the data to find the line of best fit and the correlation coefficient. c) Describe the relationship between these sprint times and throwing distances. d) State which data points could be identified as outliers, and explain why you chose them. e) Remove the outliers and repeat the regression analysis. Determine the line of best fit and the correlation coefficient for this smaller sample. f) What might the coach conclude from this analysis? What limits the predictions he could make? g) Use the two regression equations from parts b) and e) to estimate the throwing distance for a player whose sprint time is 6.50 s. 15. A biologist records the following data on the growth of a cell culture: Time (hours) Cell Count a) Draw a scatter plot for these data. b) Try two different non-linear regression models for the data, and record the regression equation and coefficient of determination for each model. c) Which model is better? Explain why. d) Predict the size of the cell culture at 8.5 h. e) Estimate the time at which the number of cells will reach

5 One Variable and Two Variable Statistics Answer Section SHORT ANSWER 1. You and your financial planner have interpreted the word typical differently. The planner was referring to the mode of the minimum investment amounts, while you calculated the mean. 2. mean 16, median 15, mode, mean 22, median 25, mode, mean 12, median 14.5, mode a) mean 44.1, median 49, mode 53 b) The mean indicates that the arithmetic average of the group s ages is about 44. The median indicates that half of the friends are under 49 and the other half are over 49. The mode indicates that 53 is the most common age in the group, but this information is not significant since the group is so small. c) Since the median is higher than the mean, the group s ages must be unevenly distributed. 6. median $30, first quartile $20, third quartile $50 7. range $110, interquartile range $30 8. (93, 19) 9. The exponential regression y = 2.5(3) x has r 2 = The cubic regression y = x has r 2 = 1. PROBLEM a) x = 9 = 13 Next, arrange the data in ascending order: 3, 9, 10, 12, 13, 13, 13, 21, 23 Clearly, the middle value and the most common value are both 13, so this value is both the median and the mode. b) The machine could have been shut down for maintenance or repair for most of the day on which it was used for only 3 h. If so, the hours of use on that day would not reflect the demand for the machine. 1

6 12. a) Since there are 21 cities listed, the median is simply the 11th highest value in the set of data: The first quartile is the midpoint between the fifth and sixth lowest values, so Q 1 = Similarly, the third quartile is the midpoint between the fifth and sixth highest values, so Q 3 = The median and quartiles can be calculated with a graphing calculator by entering the data into a list and then using the 1-Var Stats function from the STAT CALC menu. In a spreadsheet, you can use the MEDIAN and QUARTILE functions. b) The range is the highest value minus the lowest one: = The interquartile range is Q3 Q1 = c) The 21 cities can be considered a sample of all the cities in Canada. Therefore, use the sample version of the formulas for the mean, standard deviation, and variance. On a graphing calculator, the 1-VAR Stats function will calculate both x and s. In Microsoft Excel, you can use the AVERAGE, STDEV, and VAR functions to calculate x, s, and s 2, respectively. In Corel Quattro The resulting values are x = , s = , and s 2 = d) For Windsor, z = = e) For Toronto, z = =

7 13. a) The amount of sleep is the independent variable, so it is shown on the x-axis. b) The correlation coefficient can be calculated using the formula below, the linreg(ax+b) instruction on a graphing calculator, or the CORREL function in a spreadsheet. Ê ˆ n (xy) Ê ˆ x Ë Á y Ë Á r = È È n x 2 Ê 2 ˆ x Ë Á ÎÍ n y 2 Ê 2 ˆ y Ë Á ÎÍ = 0.76 c) There is a strong positive correlation between amounts of sleep and average marks for these students. However, this correlation does not prove that getting more sleep causes higher marks. 3

8 14. a) b) The linear regression can be done with a graphing calculator, a spreadsheet, or Fathom. As shown in the spreadsheet screen above, the equation for the line of best fit is y = 2.48x with r = c) There is a moderate negative linear correlation between the sprint times and throwing distances. d) On the scatter plot, points (7.55, 40) and (7.75, 26) appear to be outliers since they are somewhat removed from the rest of the data. e) If the two possible outliers are removed, the line of best fit becomes y = 2.17x with r = f) There appears to be a negative linear correlation between the sprint times and throwing distances. In other words, the faster runners tend to throw the ball farther. However, a sample of 12 is too small to make any reliable predictions, and the coach does not have enough data to determine whether the possible outliers really are outliers. The correlation between sprint times and throwing distances may, in fact, be only moderate. 4

9 g) Using the regression with the possible outliers, y = 2.48(6.50) = 35.5 m Using the regression without the possible outliers, y = 2.17(6.50) = 34.9 m 5

10 15. a) b) Answers may vary. Exponential, quadratic, cubic, and power regressions are shown below. c) The exponential regression model has the highest coefficient of determination. In fact, its value for r 2 is very close to 1, indicating an almost perfect fit to the data. d) Using the exponential regression equation, y = 100.1(1.994) 8.5 = After 8.5 h, there will be approximately cells. e) Using the exponential regression equation, 100.1(1.994) x = (1.994) x = x log(1.994) = log x = = 10.0 There will be cells in about 10.0 h. 6