Math 3339 Homework 2 (Chapter 2, 9.1 & 9.2)

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1 Math 3339 Homework 2 (Chapter 2, 9.1 & 9.2) Name: PeopleSoft ID: Instructions: Homework will NOT be accepted through or in person. Homework must be submitted through CourseWare BEFORE the deadline. Print out this file and complete the problems or you can complete it using your computer. Use blue or black ink or a dark pencil if completing this by hand. Write your solutions in the space provided. You must show all work for full credit. Submit this assignment at under Assignments" and choose hw2. Total possible points: Section a. Problem 1 b. Problem 2 c. Problem 3 d. Problem 4 e. Problem 8 a. Mean = 87.12/50 = ; Median is the average of the 25 th and 26 th value Median = ( )/2 = 1.53 b. > median(times) [1] 1.53 > mean(times) [1] c. 60 th percentile: 50* = 30.5; the 60 th percentile is the mean of the 30 th and 31 st number ( )/2 = 1.79 d. Using R studio: > quantile(times,0.6,type=5) This depends on what type of algorithm we designate. 60% 1.79 e. > Times[40]=232 > mean(times) [1] > median(times) [1] 1.53

2 2. Using R Studio and the data set precip determine the following. Precip - The average amount of precipitation (rainfall) in inches for each of 70 United States (and Puerto Rico) cities. Hint: the variable precip is already downloaded into R studio. To determine how to get the following see R studio quick reference guide. a. Mean b. Median c. Standard deviation d. Five number summary e. IQR (Determine if there are outliers). f. Histogram g. Boxplot h. Describe the shape of the distribution, i.e., skewed right, skewed left, symmetric, bimodal. a. > mean(precip) [1] b. median(precip) [1] 36.6 c. > sd(precip) [1] d. > fivenum(precip) Phoenix Milwaukee Pittsburg Providence Mobile e. IQR = = 13.7, any observations outside the interval ( *13.7, *13.7) = (8.55, 63.35) is an outlier Yes there are outliers: Phoenix Reno Albuquerque El Paso Mobile f. hist(precip) g. boxplot(precip,horizontal = T)

3 h. Shape: Somewhat skewed left 3. Section 2.3.4, Problem 3. > sd(times) [1] > var(times) [1]

4 4. *A sample of 20 glass bottles of a particular type was selected, and the internal pressure strength of each bottle was determined. Consider the following partial sample information: Median = 202.2, Q1 = 196, Q3 = Three smallest observations: Three largest observations: a. Are there any outliers in this sample? If so give the values. b. Construct a boxplot that shows outliers and comment on any interesting features. a. IQR = = 20.8; Any observation outside the interval ( *20.8, *20.8) = (164.8, 248) is an outlier. Outliers: and b. This appears to be skewed right

5 5. From: Business Statistics in Practice, 7 th edition, Bowerman, O Connell and Murphree Consider three stock funds, which we will call Stock Funds 1, 2, and 3. Suppose that Stock Fund 1 has a mean yearly return of percent with a standard deviation of percent, Stock Fund 2 has a mean yearly return of 13 percent with a standard deviation of 9.36 percent, and Stock Fund 3 has a mean yearly return of percent with a standard deviation of 41.6 percent. Give a sentence or two to answer the question Which fund is riskier? Hint: Determine the coefficient of variation for all three stock funds. Fund 1: cv = 41.96/10.93 = Fund 2: cv = 9.36/13 = 0.72 Fund 3: cv = 41.6/34.45 = Fund 1 has a higher cv, this appears to be more riskier than the other funds.

6 6. Below is a stem-plot of the birth weights of male babies born to the smoking group. The stems are in units of kg. The decimal point is at the a. Find the median birth weight. b. Find the mean birth weight. c. Find the sample standard deviation of the birth weight. d. Which measurement would be best to use for measuring the center? Justify your answer. a. There are 27 in this group, so the 13 th value is the median, Median = 3.6 b. Mean = 99.1/27 = c. SD = d. Since this is skewed to the right, the mean is slightly higher than the median. Thus the median might be a better use for the center.

7 , Problem 6 Ozone shows outliers and are very skewed right. Solar.R is skewed left. Wind is symmetric with outliers. Temp is somewhat symmetric.

8 8. In R Studio use the data cars to determine the following. Hint: The data set is already in R studio use the quick reference guide to determine the following. Description: The data gives the speed of cars and the distances taken to stop. Note that the data were recorded in the 1920s. Format A data frame with 50 observations on 2 variables. speed numeric Speed (mph) dist numeric Stopping distance (ft) a. Give a scatter plot of the data. Determine the form, direction and strength of the relationship between speed and stopping distance (dist). b. Determine the LSRL for predicting stopping distance based on speed of the car. c. Interpret the slope of this LSRL equation. d. Determine the correlation. Give an interpretation of the correlation. e. Determine the coefficient of determination, R 2. Give an interpretation of R 2. f. One of the cars was going 25 mph and had a stopping distance of 85 feet. Determine the residual of this car. a. Postive, linear, somewhat strong relationship. b. The following is from R-studio > summary(lm(dist~speed)) Call: lm(formula = dist ~ speed) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) * speed e-12 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 48 degrees of freedom

9 Multiple R-squared: , Adjusted R-squared: F-statistic: on 1 and 48 DF, p-value: 1.49e-12 LSRL: yy = xx c. Interpret of the slope ββ 1 = , for each additional mph of speed, the stopping distance is estimated to increase by about 4 ft. d. From R: > cor(speed,dist) [1] This is a strong positive relationship between speed and stopping distance. e. R 2 = , About 65% of the variation in the stopping distance can be explained by this least squares equation. f. For 25 mph, the predicted y = (25) = ft. Residual = observed y predicted y = = , Problem 2 > primates.lm Call: lm(formula = log(brain) ~ log(body), data = primates) Coefficients: (Intercept) log(body) yy = xx 10. Answer True or False for the following statements. If false, justify what would make the statement true. a. If the least-squares equation relating the independent variable x and the dependent variable y for a given problem is y = 2x+5, then an increase of 1 unit in x is associated with an increase of 2 units in y. b. If your computed correlation coefficient is r = +1.2, then you have better than a perfect positive correlation. c. A student might expect that there is a positive correlation between the age of his or her computer and its resale value. d. In simple regression analysis, if the slope of the line is positive, then there is a positive correlation between the dependent variable y and the independent variable x. e. If there is no correlation between the independent and dependent variables, then the value of the correlation coefficient must be 1. a. Ture b. False, because the correlation coefficient has to be between -1 and +1. c. False, this will be a negative correlation. d. True e. False, if there is no correlation then the correlation coefficient is near zero.