Questions and Answers of Descriptive Statistics

A biased die has the probability distribution * 1 2 3 4 S 6 P(X = x) 0.1 0.1 0.1 0.2 0.4 0.1 Find E(X) and E(X).AppendixLO1
The random variable X has a probability function P(X = x)=-2,3,6 0 all other values a Construct a table giving the probability distribution of X. b Work out E(X) and E(X2). c State with a reason
Two coins are tossed 50 times. The number of heads is noted each time. a Construct a probability distribution table for the number of heads when the two coins are tossed once, assuming that the two
The random variable X has the following probability distribution. x 1 2 3 4 5 P(X-x) 0.1 a b 0.2 0.1 Given E(X) = 2.9 find the value of a and the value of b.AppendixLO1
A fair spinner with equal sections numbered 1 to 5 is thrown 500 times. Work out how many times it can be expected to land on 3.AppendixLO1
For the following probability distribution x -1 0 1 2 3 P(X = x) 43 IS 13 55 13 a write down E(X), b find Var(X).AppendixLO1
Find the expectation and variance of each of the following distributions of X. a x P(X = x) 1 2 3 x P(X = x) -1 IN - 0 1 e -14 x P(X = x) -2 -1 216 1 16 AppendixLO1
Given that Y is the score when a single unbiased die is rolled, find E(Y) and Var(Y).AppendixLO1
Two fair cubical dice are rolled and S is the sum of their scores. a Find the distribution of S. b Find E(S). c Find Var(S).AppendixLO1
Two fair tetrahedral (four-sided) dice are rolled and D is the difference between their scores. a Find the distribution of D and show that P(D = 3) = b Find E(D). c Find Var(D).AppendixLO1
A fair coin is tossed repeatedly until a head appears or three tosses have been made. The random variable T represents the number of tosses of the coin. a Show that the distribution of T is t 1 2 3
The random variable X has the following distribution: x 1 2 3 P(X = x) a b a where a and b are constants. a Write down E(X). b Given that Var(X) = 0.75, find the values of a and b.AppendixLO1
E(X) = 4, Var(X) = 10 Find a E(2X), AppendixLO1
E(X) = 2, Var(X) = 6 Find a E(3X), e Var (3X), b Var (2X). b E(3X + 1), C E(X-1), d E(4-2X), f Var (3x+1), g Var (X-1).AppendixLO1
The random variable X has a mean of 3 and a variance of 9. Find a E(2X + 1), b E(2 + X), Var(2X + 1), d Var(2 + X).AppendixLO1
The random variable X has a mean and standard deviation . Find, in terms of and o u a E(4X), b E(2X + 2), C E(2X-2), d Var(2X + 2), e Var(2X-2).AppendixLO1
The random variable Y has mean 2 and variance AppendixLO1
Find: a E(3Y+1), b E(2-3Y), c Var(3Y + 1), d Var(2-3), e E(Y), f E[(Y 1)(Y+1)].AppendixLO1
A fair spinner is made from the disc in the diagram and the random variable X represents the number it lands on after being spun. a Write down the distribution of X. c Find Var(X). e Find Var(3X-1).
The discrete variable X has the probability distribution 0 12 x -1 P(X = x) 0.2 0.5 0.2 0.1 a Find E(X), b Find Var (X), c Find E(X + 1), d Find Var(x + 1).AppendixLO1
X is a discrete uniform distribution over the numbers 1, 2, 3, 4 and 5, Work out the expectation and variance of X.AppendixLO1
Seven similar balls are placed in a bag. The balls have the numbers 1 to 7 on them. A ball is drawn out of the bag. The variable X represents the number on the ball. a Find E(X). b Work out
A fair die is thrown once and the random variable X represents the value on the upper face. a Find the expectation and variance of X. b Calculate the probability that X is within one standard
A card is selected at random from a pack of cards containing the even numbers 2, 4, 6, ..., 20. The variable X represents the number on the card. a Find P(X > 15). b Find the expectation and variance
Repeat Question 4 for the odd numbers 1, 3, 5, ..., 19, AppendixLO1
A straight line is drawn on a piece of paper. The line is divided into four equal lengths and the segments are marked 1, 2, 3 and 4. In a party game a person is blindfolded and asked to mark a point
In a fairground game the spinner shown is used. It cost 5p to have a go on the spinner. The spinner is spun and the number of pence shown is returned to the contestant.If X is the number which comes
State whether each of the following variables is qualitative or quantitative. a Height of a tree. b Colour of car. c Time waiting in a queue. d Shoe size. e Name of pupils in a class.AppendixLO1
State whether each of the following quantitative variables is continuous or discrete. a Shoe size. b Length of leaf. c Number of people on a bus. d Weight of sugar. e Time required to run 100 m. f
Explain why a 'Type of tree' is a qualitative variable. b 'The number of pupils in a class' is a discrete quantitative variable. c "The weight of a collie dog' is a continuous quantitative
The nurse at a health centre records the heights, h cm, to the nearest cm, of a group of boys in the same school age group. The frequency table shows the results. Cumulative h Frequency (f) frequency
The distribution of the lifetimes of torch batteries is shown in the grouped frequency table below. Lifetime Frequency Cumulative frequency (nearest 0.1 of an hour) 5.0-5.9 5 6.0-6.9 8 7.0-7.9 10
The distribution of the weights of two-month-old piglets is shown in the grouped frequency table below. Weight (kg) Frequency Cumulative frequency 1.2-1.3 8 1.3-1.4 28 1.4-1.5 32 1.5-1.6 22 a Write
Write down which of the following statements are true. a The weight of apples is discrete data. b The number of apples on the trees in an orchard is discrete data. c The amount of time it takes a
Meryl collected wild mushrooms every day for a week. When she got home each day she weighed them to the nearest 100 g. The weights are shown below. 500 700 400 300 900 700 700 a Write down the mode
Joe collects six pieces of data x1, x2, x3, x4, x5 and x. He works out that x is 256.2. a Calculate the mean for these data. He collects another piece of data. It is 52. b Write down the effect this
A small workshop records how long it takes, in minutes, for each of their workers to make a certain item. The times are shown in the table. Worker A B C D E F G HI Time in minutes 7 12 10 8 6 8 5 26
A farmer keeps a record of the weekly milk yield of his herd of seven cows over a period of six months. He finds that the mean yield is 24 litres. He buys another cow that he is told will produce 28
A clothes retailer has two shops in the town of Field-gate. Shop A employs 15 people and shop B employs 22 people. The mean number of days of sickness in a year taken by the employees in shop A is
The rainfall in a certain seaside holiday resort was measured, in millimetres, every week for ten weeks. The hours of sunshine were also recorded. The data are shown in the table. Rainfall (mm) 0 1 2
The mean marks for a statistics exam were worked out for three classes. Class 1 had 12 students with a mean mark of 78%. Class 2 had 16 students with a mean mark of 84%. Class 3 had 18 students with
The marks scored in a multiple choice statistics test by a class of students are: 5 9 6 10 9 8 5 5 7 9 7 8 6 10 10 7 9 6 9 7 7 7 8 6 9 7 8 6 7 8 7 9 8 5 a Draw a frequency distribution table for
The table shows the number of eggs laid in 25 blackbirds' nests. Number of eggs 01 2 3 4 5 6 7 Number of nests 0 0 0 1 3 9 8 4 Using your knowledge of measures of location decide what number of eggs
The table shows the frequency distribution for Number of Frequency the number of petals in the flowers of a group of celandines. petals (A) a Work out how many celandines were in the group. 5 8 b
The frequency table shows the number of breakdowns,b, per month recorded by a road haulage firm over a certain period of time. Breakdowns b Frequency f Cumulative frequency 0 8 8 1 11 19 2 12 31 3 3
A company makes school blazers in eight sizes. The first four sizes cost 48. The next three sizes cost 60 and the largest size costs 76.80. Write down, with a reason which of the mean, mode, or
A hotel is worried about the reliability of its lift. It keeps a weekly record of the number of times it breaks down over a period of 26 weeks. Number of breakdowns 0-1 Frequency of breakdowns (f)
The weekly wages (to the nearest ) of the production line workers in a small factory is shown in the table. a Write down the modal class. Weekly Number of wage workers,f, 175-225 4 b Calculate an
The noise levels at 30 locations near an outdoor concert venue were measured to the nearest decibel. The data collected is shown in the grouped frequency table. Noise (decibels) 65-69 70-74 75-79
DIY store A considered that it was good at employing older workers. A rival store B disagreed and said that it was better. The two stores produced a frequency table of their workers' ages. The table
The speeds of vehicles passing a checkpoint were measured over a period of one hour, to the nearest mph. The data collected is shown in the grouped frequency table.Speed (mph) 21-30 31-40 41-50 51-60
Calculate the mean of the following data set (x) using the coding y 50 80 30 110 90 70 60 = x 10 = x=3 7 AppendixLO1
Find the mean of the following data set (x) using the coding y 52 73 31 73 80 17 24 38 AppendixLO1
a Calculate the mean of 1, 2, 3, 4, 5 and 6. Using your answer to a: b Write down the mean of: i 2, 4, 6, 8, 10 and 12, ii 10, 20, 30, 40, 50 and 60, iii 12, 22, 32, 42, 52 and 62. 4. The coded mean
The grouped frequency table shows the age (a years) at which a sample of 100 women had their first child. Age of Women Frequency (a years) (f) Mid-point (x) y = *-2* x-14 11-21 11 21-27 24 27-31 27
The following figures give the number of children injured on English roads each month for a certain period of seven months. 55 72 50 66 50 47 38 a Write down the modal number of injuries. c Calculate
The mean Science mark for one group of eight students is 65. The mean mark for a second group of 12 students is 72. Calculate the mean mark for the combined group of 20 students.AppendixLO1
A computer operator transfers an hourly wage list from a paper copy to her computer. The data transferred is given below: 5.50 6.10 7.80 6.10 9.20 91.00 11.30 a Find the mean, mode and median of
A piece of data was collected from each of nine people. The data was found to have a mean value of 35.5. One of the nine people had given a value of 42 instead of a value of 32. a Write down the
On a particular day in the year 2007, the prices (x) of six shares were as follows: 807 967 727 167 207 767 Use the coding y = *7to work out the mean value of the shares.AppendixLO1
The coded mean of employee's annual earnings (Ex) for a store is 18. The code used was x720 Work out the uncoded mean earnings.AppendixLO1
Different teachers using different methods taught two groups of students. Both groups of students sat the same examination at the end of the course. The students' marks are shown in the grouped
The table summarises the distances travelled by 150 students to college each day. a Use interpolation to calculate the median distance for these data. The mid-point is x and the corresponding
Chloe enjoys playing computer games. Her parents think that she spends too much time playing games. Chloe decides to keep a record of the amount of time she plays computer games each day for 50 days.
The lifetimes of 80 batteries, to the nearest hour, is shown in the table opposite. a Write down the modal class for the lifetime of the batteries. b Use interpolation to find the median lifetime of
A linear model fit to RFFT scores and age for 500 randomly sampled individuals from the PREVEND data has equation \(\widehat{R F F T}=137.55-1.26(\) Age).(a) Interpret the slope and intercept values
According to a report on sleep deprivation by the Centers for Disease Control and Prevention, the proportion of California residents who reported insufficient rest or sleep during each of the
Cardiopulmonary resuscitation (CPR) is a procedure commonly used on individuals suffering a heart attack when other emergency resources are not available. This procedure is helpful in maintaining
Red-green colorblindness is a commonly inherited form of colorblindness; the gene involved is transmitted on the X chromosome in a recessive manner. If a male inherits an affected X chromosome, he is
The most common form of colorblindness is a recessive, sex-linked hereditary condition caused by a defect on the X chromosome. Females are XX, while males are XY. Individuals inherit one chromosome
Sex linkage refers to the inheritance pattern that results from a mutation occurring on a gene located on a sex chromosome. A classic example of a sex-linked trait in humans is red-green color
The following are cutoff values for the upper 5% of a \(t\)-distribution with either degrees of freedom 10, 50 , or 100: \(2.23,1.98\), and 2.01. Identify which value belongs to which distribution
Researchers interested in lead exposure due to car exhaust sampled the blood of 52 police officers subjected to constant inhalation of automobile exhaust fumes while working traffic enforcement in a
Researchers collected a simple random sample of 36 children who had been identified as gifted in a large city. The following histograms show the distributions of the IQ scores of mothers and fathers
Suppose that you are interested in determining whether exposure to the organochloride DDT, which has been used extensively as an insecticide for many years, is associated with breast cancer in women.
It is hypothesized that the blue-green color of the eggshells of many avian species represents an informational signal as to the health of the female that laid the eggs. To investigate this
A diamond's price is determined by various measures of quality, including carat weight. The price of diamonds increases as carat weight increases. While the difference between the size of a 0.99
In a study examining 131 collared flycatcher eggs, researchers measured various characteristics in order to study their relationship to egg size (assayed as egg volume, in \(\mathrm{mm}^{3}\) ).
The table provides summary statistics on highway fuel economy of cars manufactured in 2012. Use these statistics to calculate a \(98 \%\) confidence interval for the difference between average
While placebo treatment can influence subjective symptoms, it is typically believed that patient response to placebo requires concealment or deception; in other words, a patient must believe that
In Exercises 5.29 and 5.31 we compared the effects of two types of feed at a time. A better analysis would first consider all feed types at once: casein, horsebean, linseed, meat meal, soybean, and
The China Health and Nutrition Survey aims to examine the effects of the health, nutrition, and family planning policies and programs implemented by national and local governments. It, for example,
Guppies are small, brightly colored tropical fish often seen in freshwater fish aquariums. A study was conducted in 147 male guppies to examine the relationship between coloration and heterozygosity;
Exercise 6.10 showed a plot of length versus height for 147 male guppies with a least squares regression line.(a) Identify two points that have relatively high leverage and discuss whether these
The residual plots below are for the linear model fit in Exercise 6.10 predicting length from height for 147 male guppies.(a) From a plot of residual values versus predicted values, are the
Multilocus heterozygosity (MLH) is reflective of genetic quality; according to sexual selection research, it is thought that sexual ornamentation functions as a visual indicator of fitness. By
Exercise 5.24 introduced data on the price of diamonds based on whether a diamond is 0.99 carats or 1 carat. Based on the summary statistics, write an estimated model equation predicting price from a
Exercise 5.28 introduced data from an analysis investigating whether hatch weights between transgenic and non-transgenic chicks differ.(a) Write an estimated least squares regression line for a model
Exercise 6.20 introduced a linear model for predicting relative orange area from proportion of loci that are heterozygous (MLH). Relative orange area refers to the percentage of the body that is
The following regression output is for predicting RFFT score of 500 randomly sampled individuals from the PREVEND data based on age (years).(a) Do these data provide statistically significant
Exercise 5.28 introduced data from an analysis investigating whether hatch weights between transgenic and non-transgenic chicks differ. Based on the results from conducting the twogroup test, explain
Exercise 6.27 introduces data from a random sample of 170 married couples in Britain, where both partners' ages are below 65 years, and fits a model predicting wife's age from husband's age. Wife's
The relationship between length and height for 147 male guppies was introduced in Exercise 6.10, which used the summary statistics to calculate the equation of the least squares line for length as a
The summary table below shows the results of a multiple regression model of RFFT score versus statin use and age.In a clinical setting, the interpretive focus lies on reporting the nature of the
Can the results of the analysis in Exercise 7.1 be used to conclude that as one ages, one's cognitive function (as measured by RFFT score) declines? Explain your answer.Data From Exercise 7.1 7.1
Wolbachia is a microbial symbiont estimated to be hosted by about \(40 \%\) of all arthropod species, transmitted primarily from females to their offspring through the eggs. Researchers conducted a
A model was fit for a random sample of 100 low birth weight infants born in two teaching hospitals in Boston, Massachusetts, regressing birthweight on the predictors gestational age and toxemia
One research question of public health interest is to determine the extent to which high blood pressure is a genetic phenomenon. In 20 families, the systolic blood pressure of the mother, father, and
Exercise 7.4 introduced a study about Wolbachia and reproductive success in a wasp host. The following table shows the model coefficients for a model predicting the number of eggs laid over a
A study was conducted at a university outpatient primary care clinic in Switzerland to identify factors associated with difficult doctor-patient encounters. The data consist of 527 patient

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