BIVARIATE ANALYSIS AND INFERENTIAL STATISTICS


ASSESSMENT TWO PART TWO: BIVARIATE ANALYSIS AND INFERENTIAL STATISTICS
TOTAL OF UNIT MARK: 20% 
DUE DATE: 11:59pm WST Friday 21st October 2016 (Submit via Blackboard)

This assignment requires you to analyse and interpret the data in the table below (i.e. the data from Assessment Two Part One) by completing a series of set tasks. The data was collected from 22 participants, each of whom had their resting heart rate measured and recorded after they had been lying down for 15 minutes. While resting heart rate is generally lower the fitter you are, it can also be influenced by other factors. Hence the participants also had their height and weight measured and recorded, and provided information about their age, gender and smoking status.

The variables in the data are:
Resting Heart Rate The resting heart rate of the participant in beats per minute (bpm)
Age The age of the participant
Gender Whether the participant is male or female
Height The height of the participant in centimetres (cm)
Weight The weight of the participant in kilograms (kg)
Smoking Status Whether the participant is a non-smoker, or whether they smoke
occasionally or regularly

The data relating to these variables for the 22 participants is as follows:
Resting Heart Rate (bpm) Age Gender Height (cm) Weight (kg) Smoking Status
62 18 Male 178 80.3 Non-smoker
70 25 Female 165 57.2 Occasionally
63 63 Male 172.5 75.4 Non-smoker
59 47 Male 185 77.3 Non-smoker
72 50 Female 167 63.2 Occasionally
68 28 Female 155.5 55.1 Non-smoker
69 33 Male 166 65.2 Occasionally
80 39 Female 170 63.1 Regularly
64 38 Male 175 70.3 Non-smoker
70 45 Male 181.5 92.6 Occasionally
75 19 Female 167 57.3 Regularly
67 24 Female 152 53.2 Non-smoker
73 65 Male 165.5 64.3 Regularly
70 58 Female 180.5 62.1 Non-smoker
75 42 Male 185 93.2 Regularly
72 39 Female 173 65.4 Occasionally
71 21 Female 174 82.1 Non-smoker
65 22 Male 190.5 101.4 Non-smoker
81 61 Female 159 58.9 Regularly
57 38 Male 178.5 83.5 Non-smoker
62 53 Female 154 55.3 Non-smoker
67 27 Female 172 64.5 Non-smoker
Use the information provided on the previous page to complete the following tasks (show working where you can, as in some cases this may allow you to obtain partial marks for incorrect solutions, and give all answers to 2 decimal places unless otherwise specified):


1. Display the Gender and Smoking Status variables in a cross-tabulation, and use it to determine what percentage of the male participants smoke.               (5 marks)

2. Create a scatter plot to display the relationship between the Resting Heart Rate and Age variables, then determine the correlation coefficient and comment on the relationship (if any) between the variables.         (5 marks)

3. A follow-up to this study is going to involve the participants recording their heart rate during moderate to intense physical activity. This should be about 70% of the person’s maximum heart rate (i.e. the upper limit of how fast their heart can beat), which can be estimated by subtracting their age from 220. Use this information to complete the following:
a. Write an equation giving a person’s approximate ‘target’ heart rate during moderate to intense physical activity in terms of their estimated maximum heart rate, using T to represent the Target Heart Rate variable and M to represent the Maximum Heart Rate variable. (1 mark)

b. Write an equation giving a person’s estimated maximum heart rate (M) in terms of their age, using A to represent the Age variable. (1 mark)

c. Use the two equations you have created previously to write an equation in slope-intercept form giving a person’s approximate target heart rate (T) during moderate to intense physical activity in terms of their age (A).             (2 marks)


d. Use your equation in c. to determine an approximate target heart rate for the participant who is 18, rounded to the nearest whole number.             (2 marks)

e. Use your equation in c. to determine what the age of a person with a target heart rate of 133bpm should be.             (3 marks)

f. Your equations in a., b. and c. all define what kind of functions? (1 mark)


4. The histogram and box plot below display the distribution of the Resting Heart Rate variable for the participants:




Furthermore, the skewness of this sample is 0.098, while the kurtosis is -0.185. Use this information to complete the following:
a. Explain how you can use the histogram, box plot, skewness and kurtosis to conclude that the Resting Heart Rate variable is normally distributed in this sample.     (4 marks)

b. Given that the sample is normally distributed and has a mean of 68.73bpm and standard deviation of 6.19bpm, determine the probability that a person selected at random from the sample has a Resting Heart Rate greater than 70bpm.     (3 marks)

The data from these participants is going to be used to make predictions for a larger population, and as such can be considered a sample. Use this information to complete the remaining questions:
5. Complete the following:    
a. Calculate the sample standard deviation of the Weight variable, rounded to two decimal places.     (2 marks)

b. This sample standard deviation can be used to estimate the standard deviation for a larger population; what term do we use to refer to such a sample statistic?       (1 mark)


c. The 95% confidence interval for the Age variable is (32.23, 45.50). Explain how you can use it to determine the sample mean of the variable, and determine this sample mean rounded to the nearest whole number.                   (2 marks)

d. Explain how you would interpret the 95% confidence interval provided in c. in terms of the population mean for the Age variable.       (1 mark)

6. Given that the Resting Heart Rate variable is Normally distributed, determine:  
a. Null and alternative hypotheses, for testing whether the population mean for the Resting Heart Rate variable is equal to 68bpm.     (2 marks)

b. What statistical test would be best to use to test these hypotheses?       (1 mark)


c. If the hypothesis test above is conducted with a level of significance of 5%, and the resulting p value is 0.587, what conclusion would this lead the researcher to and why?     (2 marks)

d. What statistical test would be used to test whether the population mean of the Resting Heart Rate variable is the same for non-smokers, occasional smokers and regular smokers?   (1 mark)


e. What statistical test would be used to test whether there is an association between the Resting Heart Rate and Weight variables in the population?       (1 mark)



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