STAT170 Exam Preparation Workshop Semester

Transcription

1 Study Information STAT Exam Preparation Workshop Semester Our sample is a randomly selected group of American adults. They were measured on a number of physical characteristics (some measurements were not recorded for some of the participants) and the data was coded. The variables collected are listed below. Sex Height Wingspan Forearm Hand Forearmhand Weight BMI Sex of respondent: = Male, = Female Height measured in cm Wingspan measured in cm Forearm length measured in cm Hand length measured in cm Forearm plus hand length measured in cm Weight measured in kg BMI: = underweight = normal = overweight = obese Question a) Which of the variables above are nominal, ordinal and numerical? b) Indicate a graphical display which could be constructed from the variables listed for the following scenarios: i. Compare the heights of males and females ii. Determine whether BMI is associated with sex iii. Examine the relationship between the height and wingspan of an individual Question (c) The following data shows the ages of some of the respondents to the survey above. Use the data to answer the following questions: i. What is the mean and the standard deviation of this data set? ii. What is the median? iii. What are the upper and lower quartiles? iv. Are there any outliers? v. Sketch a boxplot based on this data.

2 Question The heights of adult American males are believed to follow a normal distribution with a mean of 7.8 cm and a standard deviation of.9 cm. a) What is the probability of finding an adult American male with a height of 7cm or more? b) In a group of adult males how many would you expect to have a height of less than 68 cm? c) A random sample of adults is selected. What is the probability that their average height is less than 7 cm? d) What is the probability that a group of males lying head to feet will end up forming a line more than metres long? Question Is the mean ratio of human forearm plus hand length to forearm length.68? The ratio of the human forearm plus hand length to their forearm length is believed to be.68. Our random sample of adults Americans had their forearm plus hand and forearm measurements taken, the ratio was then calculated for each adult and these ratios were found to have a mean ratio of.7 with a standard deviation of.6. Histogram of Forearm+ hand to forearm ratio Forearm+ hand to forearm ratio.8.8 a) Use the information given to construct and interpret a 9% confidence interval for the true mean ratio of forearm plus hand length to forearm length. b) Using the confidence interval you have just constructed, write a conclusion to the research question above.

3 Question Is there a difference between the wingspan and the height of an adult? N Mean StDev SE Mean wingspan (cm) height (cm) Difference Boxplot of wingspan (cm), height (cm) Data wing span (cm) heig ht (cm) Histogram of height (cm) 7 Histogram of wingspan (cm) 8 Histogram of Differences (with Ho and 9% t-confidence interval for the mean) _ X height (cm) wingspan (cm) Ho... Differences 7... In Da Vinci s analysis of the human body we are told that the wingspan (fingertip to fingertip with arms outstretched horizontally) of a human being is the same as their height. For our sample of American adults we measured their wingspan and their heights and obtained the following summaries. Use the summaries above with an appropriate hypothesis test to determine whether or not there is a difference in the wingspan and height of an adult. Hypothesis test

4 Question Is there an association between sex and BMI rating? A sample of adult Americans also had their weights recorded and their BMIs calculated. Of the 7 men in the sample had low to normal BMIs and the rest were considered either overweight or obese according to the BMI index. Of the females sampled 9 had low to normal BMIs and the rest were overweight or obese according to the BMI index. Use the information above and any other calculations necessary to carry out an appropriate hypothesis test to answer the research question. Hypothesis test Question 6 It is believed that 7% of American adults are either overweight or obese. Using the information in the previous question, answer the following questions. a) Estimate the proportion of American adults who are either overweight or obese? b) Do an appropriate hypothesis test to answer the research question above. Hypothesis test Are 7% of American adults either overweight or obese?

5 Question 7 Is there a difference between the forearm lengths of adult males and the forearm lengths of adult females? The output below shows a comparison between the forearm lengths of males in our sample and the forearm lengths of females in our sample. Use the output to answer the research question above. You may assume normality and equal variances for the forearm lengths of adult males and females. Two-sample T for forearm sex N Mean StDev SE Mean Difference = µ () - µ () Estimate for difference:.7 9% CI for difference: (.6,.7) T-Test of difference = (vs ): T-Value = * P-Value = * DF = * Both use Pooled StDev =.8 Hypothesis Test

6 Which measurements are useful predictors of an adult s wingspan? Question 8 We will examine the relationships between the wingspan of an adult and the length of their forearms, the length of their hands and their heights. All measurements are in centimetres. Scatterplot of wingspan (cm) vs forearm H istogram (response is wingspan (cm)) 6 wingspan (cm) forearm Residual Scatterplot of forearm versus wingspan histogram of residuals residuals versus fitted values Regression Analysis: wingspan (cm) versus forearm Model Summary S R-sq R-sq(adj) R-sq(pred) %.% 6.% Coefficients Term Coef SE Coef T-Value P-Value Constant forearm Regression Equation wingspan (cm) = forearm The output above relates to the relation between forearm length and wingspan. Does the person with the smallest forearm also have the smallest wingspan? What is the approximate wingspan of the person with the shortest forearm? Comment on the appropriateness of using the length of a person s forearm to predict their wingspan. Your comments must include references the model assumptions and the relevant plots above.

7 Question 8 continued Regression Analysis: wingspan (cm) versus hand Model Summary S R-sq R-sq(adj) R-sq(pred) % 7.9% 7.99% Coefficients Term Coef SE Coef T-Value P-Value Constant hand Regression Equation wingspan (cm) = hand Scatterplot of wingspan (cm) vs hand wingspan (cm) hand 9 The output above represents the relation between the hand length of adults and their wingspan. Draw the fitted line on the scatter plot above. The adult with the largest residual has a hand length of 7 cm and a wingspan of 9. cm. Calculate the residual. Construct and interpret a 9% confidence interval for the population slope for the relation between hand length and wingspan. You may assume that the assumption checks are reasonable. Interpret the goodness of fit statistic from the regression model for the relationship hand length and wingspan.

8 Question 8 continued Regression Analysis: wingspan (cm) versus height (cm) Model Summary S R-sq R-sq(adj) R-sq(pred) % 8.% 8.% Coefficients Term Coef SE Coef T-Value P-Value Constant height (cm).77.9 * * Regression Equation wingspan (cm) = height (cm) Fitted Line Plot wingspan (cm) = height (cm) S.997 R- Sq 8.% R- Sq(adj) 8.% wingspan (cm) height (cm) The output above represents the relation between the height of adults and their wingspan. For American adults is their height a useful predictor for their wingspan? Conduct an appropriate hypothesis test. Hypothesis Test What is the correlation coefficient? What does it measure? The sum of the residuals must be zero. What is the other requirement on the residuals when finding the Least Squares Regression line? Where possible, use the regression output given and the analyses you have made to make the following predictions. If not possible explain why not: i. Mary has a hand of length cm, what do we expect her wingspan to be? ii. Chris has a wingspan of 8 cm, what do we expect his forearm length to be? iii. Jane is 6 cm tall, what do we expect her wingspan to be? Which is the better predictor of wingspan, the length of an adult s forearm, the length of their hand or their height?