Statistics Revision Questions Nov 2016 [175 marks]

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1 Statistics Revision Questions Nov 2016 [175 marks] The distribution of rainfall in a town over 80 days is displayed on the following box-and-whisker diagram. 1a. Write down the median rainfall. 1b. Write down the minimum rainfall. 1c. Find the interquartile range. 1d. Write down the number of days the rainfall will be (i) between 43 mm and 48 mm; (ii) between 20 mm and 59 mm. A group of 100 students gave the following responses to the question of how they get to school. A χ 2 test for independence was conducted at the 5% significance level. The null hypothesis was defined as H 0 : Method of getting to school is independent of gender. 2a. Find the expected frequency for the females who use public transport to get to school. 2b. Find the χ 2 statistic.

2 2c. The χ 2 critical value is at the 5% significance level. State whether or not the null hypothesis is accepted. Give a reason for your answer. A survey investigated the relationship between the number of cleaners, n, and the amount of time, t, it takes them to clean a school. 3a. Use your graphic display calculator to write down the equation of the regression line t on n. 3b. Write down the value of the Pearson s product moment correlation coefficient, r. 3c. Use your regression equation to find the amount of time 4 cleaners take to clean the school. A study was carried out to determine whether the country chosen by students for their university studies was influenced by a person s gender. A random sample was taken. The results are shown in the following table. A χ 2 test was performed at the 1% significance level. The critical value for this test is a. State the null hypothesis. 4b. Write down the number of degrees of freedom. 4c. Write down (i) the χ 2 statistic; (ii) the associated p-value.

3 4d. State, giving a reason, whether the null hypothesis should be accepted. The cumulative frequency graph represents the speed, s, in km h 1, of 80 cars passing a speed camera. 5a. Write down the number of cars passing the camera with speed of less than or equal to 50 km h 1. 5b. Complete the following grouped frequency table for s, the speed of the cars passing the camera. 5c. Write down the mid-interval value of the 50 < s 70 interval. 5d. Use your graphic display calculator to find an estimate of (i) the mean speed of the cars passing the camera; (ii) the standard deviation of the speed of the cars passing the camera. [3 marks]

4 As part of his IB Biology field work, Barry was asked to measure the circumference of trees, in centimetres, that were growing at different distances, in metres, from a river bank. His results are summarized in the following table. 6a. State whether distance from the river bank is a continuous or discrete variable. 6b. On graph paper, draw a scatter diagram to show Barry s results. Use a scale of 1 cm to represent 5 m on the x-axis and 1 cm to represent 10 cm on the y-axis. 6c. Write down (i) the mean distance, x, of the trees from the river bank; (ii) the mean circumference, y, of the trees. 6d. Plot and label the point M( x, y) on your graph. 6e. Write down (i) the Pearson s product moment correlation coefficient, r, for Barry s results; (ii) the equation of the regression line y on x, for Barry s results. 6f. Draw the regression line y on x on your graph. 6g. Use the equation of the regression line y on x to estimate the circumference of a tree that is 40 m from the river bank.

5 A survey was conducted to determine the length of time, t, in minutes, people took to drink their coffee in a café. The information is shown in the following grouped frequency table. 7a. Write down the total number of people who were surveyed. 7b. Write down the mid-interval value for the 10 < t 15 group. 7c. Find an estimate of the mean time people took to drink their coffee. 7d. The information above has been rewritten as a cumulative frequency table. Write down the value of a and the value of b.

6 7e. This information is shown in the following cumulative frequency graph. For the people who were surveyed, use the graph to estimate (i) the time taken for the first 40 people to drink their coffee; (ii) the number of people who take less than 8 minutes to drink their coffee; (iii) the number of people who take more than 23 minutes to drink their coffee. A survey was carried out on a road to determine the number of passengers in each car (excluding the driver). The table shows the results of the survey. 8a. State whether the data is discrete or continuous. 8b. Write down the mode. 8c. Use your graphic display calculator to find (i) the mean number of passengers per car; (ii) the median number of passengers per car; (iii) the standard deviation.

7 240 cars were tested to see how far they travelled on 10 litres of fuel. The graph shows the cumulative frequency distribution of the results. 9a. Find the median distance travelled by the cars. 9b. Calculate the interquartile range of the distance travelled by the cars. 9c. Find the number of cars that travelled more than 130 km. 180 spectators at a swimming championship were asked which, of four swimming styles, was the one they preferred to watch. The results of their responses are shown in the table. A χ 2 test was conducted at the 5% significance level. 10a. Write down the null hypothesis for this test. 10b. Write down the number of degrees of freedom.

8 10c. Write down the value of χ 2 calc. 10d. The critical value, at the 5% significance level, is State, giving a reason, the conclusion to the test. The table shows the distance, in km, of eight regional railway stations from a city centre terminus and the price, in $, of a return ticket from each regional station to the terminus. 11a. Draw a scatter diagram for the above data. Use a scale of 1 cm to represent 10 km on the x-axis and 1 cm to represent $10 on the y-axis. 11b. Use your graphic display calculator to find (i) x, the mean of the distances; (ii) y, the mean of the prices. 11c. Plot and label the point M ( x, y) on your scatter diagram. Use your graphic display calculator to find 11d. (i) the product moment correlation coefficient, r; (ii) the equation of the regression line y on x. [3 marks] 11e. Draw the regression line y on x on your scatter diagram. [3 marks] 11f. A ninth regional station is 76 km from the city centre terminus. Use the equation of the regression line to estimate the price of a return ticket to the city centre terminus from this regional station. Give your answer correct to the nearest $. 11g. Give a reason why it is valid to use your regression line to estimate the price of this return ticket.

9 11h. The actual price of the return ticket is $80. Using your answer to part (f), calculate the percentage error in the estimated price of the ticket. A group of candidates sat a Chemistry examination and a Physics examination. The candidates marks in the Chemistry examination are normally distributed with a mean of 60 and a standard deviation of a. Draw a diagram that shows this information. 12b. Write down the probability that a randomly chosen candidate who sat the Chemistry examination scored at most 60 marks. 12c. Hee Jin scored 80 marks in the Chemistry examination. Find the probability that a randomly chosen candidate who sat the Chemistry examination scored more than Hee Jin. 12d. The candidates marks in the Physics examination are normally distributed with a mean of 63 and a standard deviation of 10. Hee Jin also scored 80 marks in the Physics examination. Find the probability that a randomly chosen candidate who sat the Physics examination scored less than Hee Jin. 12e. The candidates marks in the Physics examination are normally distributed with a mean of 63 and a standard deviation of 10. Hee Jin also scored 80 marks in the Physics examination. Determine whether Hee Jin s Physics mark, compared to the other candidates, is better than her mark in Chemistry. Give a reason for your answer. 12f. To obtain a grade A a candidate must be in the top 10% of the candidates who sat the Physics examination. Find the minimum possible mark to obtain a grade A. Give your answer correct to the nearest integer. [3 marks]

10 The table shows the number of bicycles owned by 50 households. Write down the value of 13a. (i) t ; (ii) w. Indicate with a tick ( 13b. ) whether the following statements are True or False. George leaves a cup of hot coffee to cool and measures its temperature every minute. His results are shown in the table below. Write down the decrease in the temperature of the coffee 14a. (i) during the first minute (between t = 0 and t =1) ; (ii) during the second minute; (iii) during the third minute. [3 marks] 14b. Assuming the pattern in the answers to part (a) continues, show that k = c. Use the seven results in the table to draw a graph that shows how the temperature of the coffee changes during the first six minutes. Use a scale of 2 cm to represent 1 minute on the horizontal axis and 1 cm to represent 10 C on the vertical axis. t 14d. The function that models the change in temperature of the coffee is y = p (2 )+ q. (i) Use the values t = 0 and y = 94 to form an equation in p and q. (ii) Use the values t =1 and y = 54 to form a second equation in p and q. Solve the equations found in part (d) to find the value of p and the value of q. 14e.

11 The graph of this function has a horizontal asymptote. 14f. Write down the equation of this asymptote. 14g. George decides to model the change in temperature of the coffee with a linear function using correlation and linear regression. Use the seven results in the table to write down (i) the correlation coefficient; (ii) the equation of the regression line y on t. 14h. Use the equation of the regression line to estimate the temperature of the coffee at t = 3. Find the percentage error in this estimate of the temperature of the coffee at t = 3. 14i. Francesca is a chef in a restaurant. She cooks eight chickens and records their masses and cooking times. The mass m of each chicken, in kg, and its cooking time t, in minutes, are shown in the following table. 15a. Draw a scatter diagram to show the relationship between the mass of a chicken and its cooking time. Use 2 cm to represent 0.5 kg on the horizontal axis and 1 cm to represent 10 minutes on the vertical axis. Write down for this set of data 15b. (i) the mean mass, m ; (ii) the mean cooking time, t. Label the point 15c. M( m, t ) on the scatter diagram. Draw the line of best fit on the scatter diagram. 15d. Using your line of best fit, estimate the cooking time, in minutes, for a 1.7 kg chicken. 15e. Write down the Pearson s product moment correlation coefficient, r. 15f. Using your value for r, comment on the correlation. 15g.

12 The cooking time of an additional 2.0 kg chicken is recorded. If the mass and cooking time of this chicken is included in the 15h. data, the correlation is weak. (i) Explain how the cooking time of this additional chicken might differ from that of the other eight chickens. (ii) Explain how a new line of best fit might differ from that drawn in part (d). The cumulative frequency graph shows the heights, in cm, of 80 young trees. Write down the median height of the trees. 16a. 16b. Write down the 75 th percentile. Find the interquartile range. 16c. Estimate the number of trees that are more than 40 cm in height. 16d.

13 The weights, in kg, of 60 adolescent females were collected and are summarized in the box and whisker diagram shown below. Write down the median weight of the females. 17a. Calculate the range. 17b. Estimate the probability that the weight of a randomly chosen female is more than 50 kg. 17c. 17d. Use the box and whisker diagram to determine if the mean weight of the females is less than the median weight. Give a reason for your answer. A market researcher surveyed men and women about their preferred holiday destination. The holiday destinations were Antigua, Barbados, Cuba, Guadeloupe and Jamaica. A χ 2 test for independence was conducted at the 5 % significance level. The χ 2 calculated value was found to be Write down the null hypothesis. 18a. Find the number of degrees of freedom for this test. 18b. Write down the critical value for this test. 18c. State the conclusion of this test. Give a reason for your decision. 18d. The number of passengers in the first ten carriages of a train is listed below. 6, 8, 6, 3, 8, 4, 8, 5, p, p The mean number of passengers per carriage is 5.6. Calculate the value of p. 19a. Find the median number of passengers per carriage. 19b. If the passengers in the eleventh carriage are also included, the mean number of passengers per carriage increases to c. Determine the number of passengers in the eleventh carriage of the train. International Baccalaureate Organization 2016

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