Outliers

Kay is studying the variables Daily Total Sunshine (x) and Daily Total Rainfall (y) from the large data set for Leeming in 2015 Kay starts with 5th May and then selects every 10th day thereafter.

State the name of the sampling technique Kay uses. Kay wants to find the regression line of y on x for these data.

Using your knowledge of the large data set, explain how Kay might need to clean these data before finding the equation of the regression line. The equation of the regression line Kay finds is y = 0.741 + 0.199x

Using your knowledge of the large data set,

state the units of the gradient of the regression line,

give an interpretation of the y-intercept of the regression line. Kay’s teacher claimed that the greater the amount of sunshine in a day the lower the amount of rain there should be.

State, giving a reason, whether the teacher’s claim is true for Kay’s data. The teacher used all the data for these variables from the large data set for Leeming in 2015, as a sample. The teacher calculated the product moment correlation coefficient for x and y to be – 0.160 In a suitable test to determine whether there is evidence to support the teacher’s claim, the p-value was 0.015

Using a 5% level of significance, state the hypotheses and conclusion for this test.

For the test in part (e) describe

the sample,

a possible population.

Ming is studying the large data set for Perth in 2015 He intended to use all the data available to find summary statistics for the Daily Mean Air Temperature, x^oC. Unfortunately, Ming selected an incorrect variable on the spreadsheet. This incorrect variable gave a mean of 5.3 and a standard deviation of 12.4

Using your knowledge of the large data set, suggest which variable Ming selected. The correct values for the Daily Mean Air Temperature are summarised as n = 184 x 2801 2. x² 44 695 4.

Calculate the mean and standard deviation for these data. One of the months from the large data set for Perth in 2015 has

for Daily Mean Air Temperature.

Suggest, giving a reason, a month these data may have come from.

Ben is studying the Daily Total Rainfall, xmm, in Leeming for 1987 He used all the data from the large data set and summarised the information in the following table.

Explain how the data will need to be cleaned before Ben can start to calculate statistics such as the mean and standard deviation.

Calculate estimates for

the mean Daily Total Rainfall,

the standard deviation of the Daily Total Rainfall. Ben suggests using the statistic calculated in part (b)(i) to estimate the annual mean Daily Total Rainfall in Leeming for 1987

Using your knowledge of the large data set,

give a reason why these data would not be suitable,

state, giving a reason, how you would expect the estimate in part (b)(i) to differ from the actual annual mean Daily Total Rainfall in Leeming for 1987

Each member of a group of 27 people was timed when completing a puzzle. The time taken, x minutes, for each member of the group was recorded. These times are summarised in the following box and whisker plot.

Find the range of the times.

Find the interquartile range of the times. For these 27 people Σ x = 607.5 and Σ x² = 17 623.25

calculate the mean time taken to complete the puzzle,

calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean.

State how many outliers Taruni would say there are in these data, giving a reason for your answer. Adam and Beth also completed the puzzle in a minutes and b minutes respectively, where a b. When their times are included with the data of the other 27 people ● the median time increases ● the mean time does not change

Suggest a possible value for a and a possible value for b, explaining how your values satisfy the above conditions.

Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).

Each of 60 students was asked to draw a 20° angle without using a protractor. The size of each angle drawn was measured. The results are summarised in the box plot below.

Find the range for these data.

Find the interquartile range for these data.

The students were then asked to draw a 70° angle. The results are summarised in the table below.

Angle, a, (degrees)	Number of students
55 a 60	6
60 a 65	15
65 a 70	13
70 a 75	11
75 a 80	8
80 a 85	7

Use linear interpolation to estimate the size of the median angle drawn. Give your answer to 1 decimal place.

Show that the lower quartile is 63°

For these data, the upper quartile is 75°, the minimum is 55° and the maximum is 84° An outlier is an observation that falls either more than 1.5 × (interquartile range) above the upper quartile or more than 1.5 × (interquartile range) below the lower quartile.

Show that there are no outliers for these data.

Draw a box plot for these data on the grid on page 3.

State which angle the students were more accurate at drawing. Give reasons for your answer.

The marks of a group of female students in a statistics test are summarised in Figure 1

Write down the mark which is exceeded by 75% of the female students.

The marks of a group of male students in the same statistics test are summarised by the stem and leaf diagram below.

Find the median and interquartile range of the marks of the male students.

An outlier is a mark that is either more than 1.5 × interquartile range above the upper quartile or more than 1.5 × interquartile range below the lower quartile.

In the space provided on Figure 1 draw a box plot to represent the marks of the male students, indicating clearly any outliers.

Compare and contrast the marks of the male and the female students.

Over a long period of time a small company recorded the amount it received in sales per month. The results are summarised below.

An outlier is an observation that falls either 1.5 × interquartile range above the upper quartile or 1.5 × interquartile range below the lower quartile.

On the graph paper below, draw a box plot to represent these data, indicating clearly any outliers.

State the skewness of the distribution of the amount of sales received. Justify your answer.

The company claims that for 75 % of the months, the amount received per month is greater than £10 000. Comment on this claim, giving a reason for your answer.

The distances travelled to work, D km, by the employees at a large company are normally

Find the probability that a randomly selected employee has a journey to work of more than 20 km.

Find the upper quartile, Q₃, of D.

Write down the lower quartile, Q₁, of D.

An outlier is defined as any value of D such that D h or D k where

Find the value of h and the value of k.

An employee is selected at random.

Find the probability that the distance travelled to work by this employee is an outlier.

In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week. The total length of calls, y minutes, for the 11 students were 17, 23, 35, 36, 51, 53, 54, 55, 60, 77, 110

Find the median and quartiles for these data.

A value that is greater than Q₃ + 1.5 × (Q₃ – Q₁) or smaller than Q₁ – 1.5 × (Q₃ – Q₁) is defined as an outlier.

Show that 110 is the only outlier.

Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier.

The value of 110 is omitted.

Show that S_yy for the remaining 10 students is 2966.9

These 10 students were each asked how many text messages, x, they sent in the same week. The values of S_xx and S_xy for these 10 students are S_xx = 3463.6 and S_xy = –18.3.

Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students.

A parent believes that a student who sends a large number of text messages will spend fewer minutes on calls.

Comment on this belief in the light of your calculation in part (e).

Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke. A doctor tested the blood of 12 patients, who claimed to smoke a packet of cigarettes a day, for cotinine. The results, in appropriate units, are shown below.

Find the mean and standard deviation of the level of cotinine in a patient’s blood.

Find the median, upper and lower quartiles of these data.

A doctor suspects that some of his patients have been smoking more than a packet of cigarettes per day. He decides to use Q₃+1.5(Q₃–Q₁) to determine if any of the cotinine results are far enough away from the upper quartile to be outliers.

Identify which patient(s) may have been smoking more than a packet of cigarettes a day. Show your working clearly.

Research suggests that cotinine levels in the blood form a skewed distribution.

Evaluate this measure and describe the skewness of these data.