Standard combined mean and SD

3 A sports club has a volleyball team and a hockey team. The heights of the 6 members of the volleyball team are summarised by \(\Sigma x = 1050\) and \(\Sigma x ^ { 2 } = 193700\), where \(x\) is the height of a member in cm . The heights of the 11 members of the hockey team are summarised by \(\Sigma y = 1991\) and \(\Sigma y ^ { 2 } = 366400\), where \(y\) is the height of a member in cm .

1. Find the mean height of all 17 members of the club.
2. Find the standard deviation of the heights of all 17 members of the club.

1 Red Street Garage has 9 used cars for sale. Fairwheel Garage has 15 used cars for sale. The mean age of the cars in Red Street Garage is 3.6 years and the standard deviation is 1.925 years. In Fairwheel Garage, \(\Sigma x = 64\) and \(\Sigma x ^ { 2 } = 352\), where \(x\) is the age of a car in years.

1. Find the mean age of all 24 cars.
2. Find the standard deviation of the ages of all 24 cars.

5 The table shows the mean and standard deviation of the weights of some turkeys and geese.

1. Find the mean weight of the 27 birds.
2. The weights of individual turkeys are denoted by \(x _ { t } \mathrm {~kg}\) and the weights of individual geese by \(x _ { g } \mathrm {~kg}\). By first finding \(\Sigma x _ { t } ^ { 2 }\) and \(\Sigma x _ { g } ^ { 2 }\), find the standard deviation of the weights of all 27 birds.

4 Farfield Travel and Lacket Travel are two travel companies which arrange tours abroad. The numbers of holidays arranged in a certain week are recorded in the table below, together with the means and standard deviations of the prices.

1. Calculate the mean price of all 51 holidays.
2. The prices of individual holidays with Farfield Travel are denoted by \(
   ) x _ { F }\( and the prices of individual holidays with Lacket Travel are denoted by \)\\( x _ { L }\). By first finding \(\Sigma x _ { F } ^ { 2 }\) and \(\Sigma x _ { L } ^ { 2 }\), find the standard deviation of the prices of all 51 holidays.

4 A group of 10 married couples and 3 single men found that the mean age \(\bar { x } _ { w }\) of the 10 women was 41.2 years and the standard deviation of the women's ages was 15.1 years. For the 13 men, the mean age \(\bar { x } _ { m }\) was 46.3 years and the standard deviation was 12.7 years.

1. Find the mean age of the whole group of 23 people.
2. The individual women's ages are denoted by \(x _ { w }\) and the individual men's ages by \(x _ { m }\). By first finding \(\Sigma x _ { w } ^ { 2 }\) and \(\Sigma x _ { m } ^ { 2 }\), find the standard deviation for the whole group.

4 Barry weighs 20 oranges and 25 lemons. For the oranges, the mean weight is 220 g and the standard deviation is 32 g . For the lemons, the mean weight is 118 g and the standard deviation is 12 g .

1. Find the mean weight of the 45 fruits.
2. The individual weights of the oranges in grams are denoted by \(x _ { o }\), and the individual weights of the lemons in grams are denoted by \(x _ { l }\). By first finding \(\Sigma x _ { o } ^ { 2 }\) and \(\Sigma x _ { l } ^ { 2 }\), find the variance of the weights of the 45 fruits.

4 The ages of a group of 12 people at an Art class have mean 48.7 years and standard deviation 7.65 years. The ages of a group of 7 people at another Art class have mean 38.1 years and standard deviation 4.2 years.

1. Find the mean age of all 19 people.
2. The individual ages in years of people in the first Art class are denoted by \(x\) and those in the second Art class by \(y\). By first finding \(\Sigma x ^ { 2 }\) and \(\Sigma y ^ { 2 }\), find the standard deviation of the ages of all 19 people.

5 The Quivers Archery club has 12 Junior members and 20 Senior members. For the Junior members, the mean age is 15.5 years and the standard deviation of the ages is 1.2 years. The ages of the Senior members are summarised by \(\Sigma y = 910\) and \(\Sigma y ^ { 2 } = 42850\), where \(y\) is the age of a Senior member in years.

1. Find the mean age of all 32 members of the club.
2. Find the standard deviation of the ages of all 32 members of the club.

3 The mean and standard deviation of 20 values of \(x\) are 60 and 4 respectively.

1. Find the values of \(\Sigma x\) and \(\Sigma x ^ { 2 }\).
   Another 10 values of \(x\) are such that their sum is 550 and the sum of their squares is 40500 .
2. Find the mean and standard deviation of all these 30 values of \(x\).

1. Two classes of students, class \(A\) and class \(B\), sat a test.

Class \(A\) has 10 students. Class \(B\) has 15 students. Each student achieved a score, \(x\), on the test and their scores are summarised in the table below. The mean score for Class \(A\) is 77 and the mean score for Class \(B\) is 61

1. Find the value of \(t\)
2. Calculate the variance of the test scores for each class. The highest score on the test was 95 and the lowest score was 45 These were each scored by students from the same class.
3. State, with a reason, which class you believe they were from. The two classes are combined into one group of 25 students.
4. 1. Find the mean test score for all 25 students.
   2. Find the variance of the test scores for all 25 students. The teacher of class \(A\) later realises that he added up the test scores for his class incorrectly. Each student's test score in class \(A\) should be increased by 3
5. Without further calculations, state, with a reason, the effect this will have on
   1. the variance of the test scores for class \(A\)
   2. the mean test score for all 25 students
   3. the variance of the test scores for all 25 students.

4 At a stall in a fair, contestants have to estimate the mass of a cake. A group of 10 people made estimates, \(m \mathrm {~kg}\), and for each person the value of \(( m - 5 )\) was recorded. The mean and standard deviation of \(( m - 5 )\) were found to be 0.74 and 0.13 respectively.

1. Write down the mean and standard deviation of \(m\). The mean and standard deviation of the estimates made by another group of 15 people were found to be 5.6 kg and 0.19 kg respectively.
2. Calculate the mean of all 25 estimates.
3. Fiona claims that if a group's estimates are more consistent, they are likely to be more accurate. Given that the true mass of the cake is 5.65 kg , comment on this claim.

10 The pre-release material contains information about the birth rate per 1000 people in different countries of the world. These countries have been classified into different regions. The table shows some data for three of these regions: the mean and standard deviation (sd) of the birth rate per 1000, and the number of countries for which data was used, n. \section*{Birth rate per 1000 by region}

1. Use the information in the table to compare and contrast the birth rate per 1000 in Africa with the birth rate per 1000 in Europe.
2. The birth rate per 1000 in Mauritius, which is in Africa, is recorded as 9.86. Use the information in the table to show that this value is an outlier.
3. Use your knowledge of the pre-release material to explain whether the value for Mauritius should be discarded.
4. The pre-release material identifies 27 countries in Oceania. Suggest a reason why only 21 values were used to calculate the mean and standard deviation.

1. Twelve observations are made of a random variable \(X\). This set of observations has mean 13 and variance \(10 \cdot 2\).

Another twelve observations of \(X\) are such that \(\sum x = 164\) and \(\sum x ^ { 2 } = 2372\).
Find the mean and the variance for all twenty-four observations.

4. A company offering a bicycle courier service within London collected data on the delivery times for a sample of jobs completed by staff at each of its two offices. The times, \(t\) minutes, for 20 deliveries handled by the company's Hammersmith office were summarised by $$\Sigma t = 427 , \text { and } \Sigma t ^ { 2 } = 11077$$

1. Find the mean and variance of the delivery times in this sample. The company's Holborn office handles more business, so the delivery times for a sample of 30 jobs handled by this office was taken. The mean and standard deviation of this sample were 18.5 minutes and 8.2 minutes respectively.
2. Find the mean and variance of the delivery times of the combined sample of 50 deliveries.

4. A student collected data on the number of text messages, \(t\), sent by 30 students in her year group in the previous week. Her results are summarised as follows: $$\Sigma t = 1039 , \quad \Sigma t ^ { 2 } = 65393 .$$

1. Calculate unbiased estimates of the mean and variance of the number of text messages sent by these students per week.
   (4 marks)
   Another student collected similar data for 20 different students and calculated unbiased estimates of the mean and variance of 32.0 and 963.4 respectively.
2. Calculate unbiased estimates of the mean and variance for the combined sample of 50 students.
   (6 marks)

16 An analysis was carried out using the Large Data Set to compare the \(\mathrm { CO } _ { 2 }\) emissions (in g/km) from 2002 and 2016. The summary statistics for the \(\mathrm { CO } _ { 2 }\) emissions, \(X\), for all cars registered as owned by either females or males is given in the table below. 16

1. Find the reduction in the mean of the \(\mathrm { CO } _ { 2 }\) emissions in 2016 compared to the mean of the CO2 emissions in 2002.
   16
2. It is claimed that the move to more electric and gas/petrol powered cars has caused the reduction in the mean \(\mathrm { CO } _ { 2 }\) emissions found in part (a). Using your knowledge of the Large Data Set, state whether you agree with this claim.
   Give a reason for your answer.
   16
3. There are 3827 data values in the Large Data Set. It is claimed that the data in the table above must have been summarised incorrectly.
   16
4. 1. Explain why this claim is being made. 16
5. (ii) State whether this claim is correct.
   Give a reason for your answer.