Descriptive Statistics. The questions that follow exemplify how the learning outcomes shown below could be examined. Work with different types of data (categorical/numerical/ordinal/discrete/continuous) in order to clarify the problem at hand. Discuss different types of studies: sample surveys, observational surveys and designed experiments. Recognise the importance of representativeness so as to avoid biased samples. Explore the distribution of data including concepts of symmetry and skewness. Use a variety of summary statistics to explore the data. Evaluate the effectiveness of different displays in representing the findings of a statistical investigation. Compare data sets using back to back stem and leaf plots. Determine the relationship between variables using scatterplots. Recognise that the correlation is a value from 1 to + 1 and that it measures the extent of linear relationship between two variables. Draw the line of best fit by eye. Make predictions based on the line of best fit. Calculate the correlation coefficient by calculator and understand that correlation does not imply causality. Use percentiles to assign relative standing. Interpret a histogram in terms of distribution of the data. Recognise standard deviation as a measure of variability. Use the calculator to calculate standard deviation. Recognise the importance of randomisation and the role of the control group in studies. Recognise bias, limitations and ethical issues of each type of study. Make decisions based on the empirical rule. 1
LCHL Descriptive Statistics Maeve was interested in finding out if you spend more time on social networking sites as you get older. She generated a random sample from censusatschool.ie. The data is shown below. It shows the responses of 30 Irish students to two of the questions How old are you and How many hours do you spend social networking per week? Age Hours spent Social Networking 13 7 15 26 16 20 17 15 13 0 17 13 13 1 15 18 14 2 14 4 12 0 15 19 16 5 16 14 13 5 17 17 12 1 15 19 17 21 14 5 12 3 16 7 16 8 12 7 16 9 15 10 16 15 15 11 15 7 15 6 2
(a) Display the data in a way which allows you to establish whether there is a relationship between age and the number of hours spent social networking. (b) Based on your display, is there a relationship between age and the number of hours spent on social networking? Explain your conclusion. (c) Based on the data find the following: correlation coefficient and an write an interpretation of the value in this context. the equation of the line of best fit and how this could be used to make predictions beyond her study. the limitations of her study (d) On completion of her research report, Maeve posed another research question. Was there evidence to suggest that if you were female there was more of a correlation between age and hours spent social networking, than if you were male? Use the data on the next page to determine whether there is evidence to support Maeve s new research question. 3
Hours spent Social Age Networking Gender Female 13 7 Female 15 26 Female 16 20 Male 17 15 Male 13 0 Male 17 13 Male 13 1 Female 15 18 Male 14 2 Male 14 4 Female 12 0 Female 15 19 Female 16 5 Male 16 14 Male 13 5 Female 17 17 Female 12 1 Male 15 19 Female 17 21 Female 14 5 Male 12 3 Male 16 7 Male 16 8 Female 12 7 Female 16 9 Male 15 10 Male 16 15 Female 15 11 Female 15 7 Female 15 6 4
Question LCHL: Descriptive Statistics To enter a particular college course, candidates must complete an aptitude test. In 2010 the mean score was 490 with a standard deviation of 100. The scores on the aptitude test are normally distributed. (a) What percentage scored between 390 and 590 on this aptitude test (b) One student scored 795 on this test. How did the student do compared to the rest of the scores? (c) The college only admits students who were among the highest 16% of the scores on this test. What score would a student need on this test to be qualified for admittance to this university? Explain your answer. (d) A student named Alice was preparing to sit the aptitude test in 2011. She heard that a score of over 650 would guarantee her a place on the course. She knew 20 people who were going to take the test. Based on the mean and standard deviation in 2010, approximately how many of the people Alice knew were likely to get a score of above 650 and secure a place on the course? Justify your answer. 5
LCHL: Descriptive Statistics On a hospital placement, a medical student was asked to record the heart rate, blood sugar levels and cortisol levels on 20 of his patients. Heart rate is measured in beats per minute (bpm), the standard measure for blood sugar levels are measured in mmol/l, the standard measurement for cortisol (the hormone which gives us an indication of stress levels) is measured in nmol/l. The student produced the following histograms based on the data gathered from 20 patients. The results for heart beat, blood sugar level and cortisol all fell within the normal ranges. (a) Describe the distribution of each histogram. (i) Heart rate: (ii) Cortisol levels: (iii) Blood sugar levels. 6
(b) (i) In which interval does the median cortisol level occur? (ii) Give an approximate value for the average heart beat per minute based on the data. (iii) The medical student gets results back from the hospital about a new patient. The cortisol level was 990nmol/L and the blood sugar level was 4.2mmol/ L. What conclusion would the medical student come to when comparing these levels to the histograms above? (c) The medical student was interested in whether there was a relationship between heart rate and cortisol levels. The raw data she collected is provided below. Heart Rate (bpm) Cortisol levels (nmol/l) 62 500 55 160 84 720 95 440 74 540 98 750 65 260 73 440 85 790 72 330 64 480 79 650 95 720 82 350 68 590 43 420 101 810 58 680 63 90 62 260 (i) Display the data in a way which allows you to examine whether there is a correlation between heart rate and cortisol levels. (ii) What kind of correlation does your display show? Justify your answer based on your graph (iii) Should a person who has a heart rate of 32bpm and a cortisol level of 790 be concerned? explain your reasoning with reference to your graph. (iv) The correlation coefficient of blood sugar levels and cortisol levels is - 0.17. Explain what this means in this context. (v) By calculating the mean and the standard deviation of the cortisol levels in the sample of 20 patients, calculate the probability that a randomly selected student will have a cortisol level of above 710nmol/L. What does this tell you about a reading of a cortisol level of above 710nmol/L? (vi) Is the reading of 90nmol/L of cortisol levels normal? Justify your answer. 7
LCOL: Descriptive Statistics. David noticed that when he drank a bottle of sports drink before going out for a run one day that his performance time improved. He set about doing an experiment to see whether drinking the sports drink increases performance when running. He recorded the times of people in his running club to complete a 5km run without drinking the sports drink and then on another day he recorded the time it took the same people to complete 5km having drank the sports drink. He recorded the information in a back to back stem and leaf plot: Without drinking the sports drink. Having drank the sports drink. 5 20 3 4 1 1 1 21 3 4 7 8 8 4 3 2 22 23 1 2 2 24 0 25 8 26 1 27 28 2 3 6 6 7 7 29 2 4 4 5 5 5 8 9 9 30 1 3 4 5 6 7 8 8 9 31 6 4 4 3 0 0 32 1 1 4 9 9 9 6 5 4 4 3 3 2 1 33 3 3 3 2 7 7 5 5 6 6 6 1 0 34 5 8 8 8 3 3 35 0 0 7 3 2 36 1 1 37 2 38 3 5 2 2 39 4 4 2 0 40 Key: 32 1 means 32.1 minutes. 8
(i) Based on the diagrams approximate the median speed without drinking the sports drink and the median speed having drank the sports drink. What does this information tell you? (ii) Compare the distributions of each of the data sets above. (iii) Is there evidence from the diagram to suggest that David s belief was true? Justify your conclusions. (iv) Make an argument against David s belief based on the two data sets. (v) After completing the experiment, David wondered how accurate his study was. He realised that he had not specified how much of the sports drink the runners should drink. He asked 20 of the runners approximately how much sports drink they had drank in millilitres and recorded it alongside their time. The results are as follows: Sports drink Time (mins) (ml) 20.3 250 21.7 100 21.8 120 24 80 28.6 300 29.4 130 29.5 300 29.9 280 32.1 300 32.1 100 33.2 80 35 220 38.3 180 20.6 100 29.2 200 29.8 250 36.1 80 29.9 120 30.9 240 30.1 280 Display the data in a way that allows you to examine the relationship between the two data sets. (vi) Is there evidence to suggest that there is a relationship between the time to complete 5km and the amount of sports drink drank before the race? (vii) The correlation coefficient for data in part is one of the following: Circle the correct correlation coefficient based on your graph above. A -0.82 B 0.13 C 0.95 D 0.6 9
JCHL: Descriptive Statistics Cian came back from a holiday in Australia and said to his younger sister Annie I think Australian teenagers are taller than Irish people. Annie who was learning about statistics in school said You have no evidence to base that on other than your own observations, I m going to investigate this further! Annie went to Censusatschool.ie and generated a random sample of the heights of 32 Irish teenagers and 32 Australian teenagers. The data is shown here: Australia Ireland Height(cm) Height (cm) 147 190 183 190 157 158 168 184 171 173 174 123 168 152 152 150 163 171 156 163 157 144 168 172 140 160 158 168 148 159 157 160 150 159 169 190 168 163 150 163 155 152 170 176 160 182 139 150 148 168 136 186 180 180 137 152 133 174 161 163 138 187 147 169 10
(a) Annie decides to display the data so that she can then analyse it in various ways. Display the data in a way that allows you to compare the data and comment on: Distribution of the data in each sample Measures of centre in each sample. The limitations in the study. Finally make a conclusion from your data to either agree with Annie s brother or not. (b) Cian s height is 163 cm. After hearing Annie s report, her brother asks What quartile is my height in? Looking at the Irish sample, identify what quartile his height is in and explain what this means in relation to the rest of the data. 11
JCHL Descriptive Statistics: In a TV programme called Chef s Mission Impossible, a celebrity chef aims to introduce healthy brain food to the Royal Navy. He creates a menu which is high in omega 3 and the crew eat three meals a day packed with nutrients which are thought to increase brain power. The crew follow the diet for three weeks. Previous to this their diet was high in fat and sugar. In order to test whether the crew s brain function has improved, their time taken in minutes to finish a task on the boat was tested. These are the times recorded before the diet change: 6.5 4.7 3.2 9.8 10.1 11.2 8.5 6.5 3.2 10.3 6.5 4.7 5.8 11.3 4.2 8.9 10.3 3.4 12.1 11.9 8.2 9.4 8.8 9.1 12.2 10.6 4.2 4.5 7.3 7.9 Here are the recorded times after the diet change: 6.5 3.2 7.8 10.3 4.8 10.8 4.1 5.3 7.2 8.3 4.1 9.9 10.3 12.5 12.6 3.1 2.9 5.3 6.1 4.8 6.2 9.3 8.5 2.9 12.7 3.1 4.6 5.1 3.7 3.3 12
(a) (i) What type of study is this? Explain your answer. (ii) What type of data is being collected? (a) Nominal (b) Discrete (c) Continuous (iii) What are the limitations of the study? (iv) Is the study open to bias? Explain your answer. (b) (i) Display the data in a way which allows you to compare it. (ii) By comparing the two data sets and comparing measures of centre and variability, analyse whether there is evidence to suggest that the diet has improved brain power? (iii) Based on the data make an argument to support the statement The introduction of a new diet did not improve the brain power of the crew of the Royal Navy. 13
JCOL: Descriptive Statistics A group of students were asked Do you get worried about your exams? They were asked to circle one of following to answer the question:,,,. The data below shows the answers from a sample of boys and girls. Boys Girls (a) How many students were in each sample? (b) Display the data in a way which allows you to compare the two samples. (c) Compare the two sets based on your display. 14