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# Statistics

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Statistics is the branch of mathematics used to collect, analyse and present data.

## What are the topics in statistics?

### Probability

In probability, we explore the idea of independent and dependent events. You will learn about calculating the chance that an event will occur using various methods such as tree and Venn diagrams and conditional and mutually exclusive events.

With Venn diagrams, you can figure out how events can happen at the same time.

U = numbers under 20

A = even numbers

B = Multiples of 3

Solutions:

Venn diagram example

Solutions:

There are 19 numbers in the entire set.

There are 3 numbers inside the intersection.

So

### Collecting data

Here, we look at sampling, including different methods of sampling and the different types of data. Some people find sampling questions the easiest to answer in an exam, but they can also be quite wordy, so it is important to pay attention – and then you will understand precisely what you are being asked to do.

There are a few different ways to categorize data. We can categorize data as quantitative or qualitative, as well as descriptive and inferential.

There are 400 students in a school, 250 girls and 150 boys. Explain how to take a stratified sample of 40 students in the school.

Solution.

We want to pick 25 girls and 15 boys (so that we have the same proportion as the entire population). One method is to assign all the girls a number from 1-250. Then using a random number generator, generate 25 numbers and then pick those 25 girls.

Repeat the same for the boys: assign them a number from 1-150. Then use your random number generator to generate 15 numbers and then pick those 15 boys.

### Measures of location and spread

We need to analyze the data we collect, and the best way to do this is by using measures of location and spread. This enables us to compare data using the following

• The mean and standard deviation.
• Using simple data analysis such as the mode and the range.
• Finding quartiles and percentiles.
• Using algebra to code this data.

On a randomly chosen day, each of the 32 students in a class recorded the time (t) in minutes to the nearest minute that it took them to get to school. Find the mean and standard deviation from the following data:

Solutions:

mean:

Default Deviation:

### Statistical distributions

A vital part of statistics is understanding the distribution of data. Distributions are essentially mathematical functions that give the probability that a function will occur. We will look at two main distributions, binomial distribution and normal distribution.

Binomial distribution applies whenever there are two mutually exclusive possible outcomes of an experiment. If an experiment with the probability of the outcome happening being p is performed n times, the probability of this outcome happening n times is:

with (also written as )

A die is tossed 10 times. The outcome of rolling 5 exhibits a binomial distribution: . Calculate .

Solution.

This is as simple as calculating and and summing them all together. Therefore:

Summing all of these together we get:

In some exams, you will have access to a formula booklet with a dedicated section for statistics. Check your exam board's website for this. The most useful formulas are the ones for binomial distribution and mutualistic probability; however, the booklet will also contain statistical tables. You might not need these if you can use a calculator, but they show probability values at significance levels on distributions.

### Hypothesis testing

Hypothesis testing involves using distribution to calculate whether or not we can say a statement ( a hypothesis) is true. In the hypothesis testing topic, we will look at conducting one-tailed and two-tailed tests and stating a null hypothesis.

A coffee shop claims that a quarter of the cakes sent to them are missing cherries on top. To test this claim, the number of cakes without cherries in a random sample of 40 is recorded. Using a 5% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a missing cherry is 0.26.

Solutions:

This is a two-tailed test, meaning we will need to look at both ends. Start at a random number on both ends.

Lower end:

P(X ≤ 6) = 0.07452108246

P(X ≤ 5) = 0.03207217407

P(X ≤ 4) = 0.01136083855

As this is a two-tailed test we want to be as close to 0.025 as possible:

P(X ≤ 4) < 0.025 < P(X ≤ 5).

Upper end:

P(X ≥ 16) = 1 - P(X ≤ 15) = 0.03703627013

P(X ≥ 17) = 1 - P(X ≤ 16) = 0.0171086868649

Again we want to be as close to 0.025 as possible so:

P(X ≥ 17) < 0.025 < P(X ≥ 16)

Our critical regions are therefore P(X ≤ 4) and P(X ≥ 17).

### Representing data

In the representing data topic, we will look at graphical methods to showcase data. These include histograms, box plots and cumulative frequency. We will also look at ways that we can find outliers in data, and how to deal with data anomalies.

If you are studying for A-level exams, some exam boards prove a large data set eg a spreadsheet containing weather data from airports in the UK and around the world. You don't have to memorize any data but what you do have to do is familiarize yourself with the different types of data it contains, and the units of these data.

## Statistics - Key takeaways

• Statistics can be broken up into many concepts.
• A lot of the concepts from Pure Mathematics are used in Statistics such as Binomial Expansion in Binomial Distribution.
• The Mathematical process in Statistics is the same as that in Pure Mathematics.

Statistics is the branch of Mathematics that deals with collecting and analysing numerical data in large quantities.

The three types are descriptive, inferential and quantitative.

Statistical measures to organise, present and summarise data in an informative manner.

Variance is a measure of spread equal to the standard deviation squared.

## Final Statistics Quiz

Question

What is probability distribution?

A probability distribution is the function that gives the individual probabilities of occurrence of different possible outcomes for an experiment.

Show question

Question

What is the sum of all the probabilities of a probability distribution?

1

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Question

Identify whether the following requires a discrete or continuous probability distribution?
The amount of rainfall in your city in March.

Continuous

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Question

Identify whether the following requires a discrete or continuous probability distribution?
The number of trophies your favourite football club will win this season.

Discrete

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Question

Identify whether the following requires a discrete or continuous probability distribution?
The number of students in the class who will pass the mathematics exam.

Discrete

Show question

Question

Identify whether the following requires a discrete or continuous probability distribution?
The weight of a newborn baby.

Continuous

Show question

Question

Identify whether the following requires a discrete or continuous probability distribution?
The number of runs Joe Root will score in his next match

Discrete

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Question

The table for a probability distribution function has only 1 entry. What is the probability of that entry?

1

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Question

For a random variable X, what does the cumulative probability, P(X ≤ 8) represent?

The probability that the outcome of the random variable is less than or equal to 8.

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Question

A biased dice with faces numbered from 1 to 6 is rolled. The number obtained is modelled as a random variable X. Given that P (X = x) = k/x, find the value of k.

k/1 + k/2 + k/3 + k/4 +k/5 + k/6 = 1

=> k = 20/49

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Question

The discrete random variable X has the probability function,

P(X = x) = 0.2, x = 0,1= a, x = 2, 3=0.3, x = 4

Find the value of a.

0.2 + 0.2 + a + a + 0.3 = 1

=> a = 0.15

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Question

An unbiased coin is tossed 3 times and X is the random variable counting the number of heads obtained. What is P (X ≤ 2)?

0.875

Show question

Question

The discrete random variable X has the probability function,

P(X = x) = kx, x = 1,2

= 0.2, x = 3,4

Find the value of k.

k + 2k + 0.2 × 2 = 1

=> k = 0.2

Show question

Question

An experiment with 10 possible outcomes results in a uniform probability distribution. What is the probability of each outcome?

0.1

Show question

Question

Which of the following is used for a discrete probability distribution?

probability mass function

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Question

Which of the following is used for a continuous probability distribution?

probability density function

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Question

There are 100 students in a Mathematics class. The random variable X represents the number of students taller than 6 feet. The cumulative probability at X=20 is 0.85. What is the probability that there are more than 20 students taller than 6 feet?

0.15

Show question

Question

Can a binomial distribution be used for the following trial: the score when a fair dice is rolled.

No

Show question

Question

Can a binomial distribution be used for the following trial: an unbiased coin is tossed.

Yes

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Question

Can a binomial distribution be used for the following trial: whether the score when a fair dice is rolled is greater than 4.

Yes

Show question

Question

What are the necessary conditions for a binomial distribution?

1) there are a fixed number of trials, n

2)there are 2 possible outcomes, success and failure

3) there is a fixed probability of success, p, for all trials

4) the trials are independent

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Question

Which of the following can be used for a binomial distribution?

probability mass function

Show question

Question

If a random variable X has the binomial distribution B(n, p), write down its probability mass function.

P(X = r) = nCr p^r (1 - p)^(n-r)

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Question

For the random variable X ~ B (5, 0.2), find P (X = 3).

0.0512

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Question

For the random variable X ~ B (5, 0.2), find P (X = 1).

0.4096

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Question

For the random variable X ~ B (5, 0.2), find P (X = 4).

0.0064

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Question

For the random variable X ~ B (9, 0.9), find P (X = 5).

0.0074

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Question

For the random variable X ~ B (9, 0.9), find P (X = 7).

0.172

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Question

For the random variable X ~ B (500, 0.4), what is the cumulative probability P (X ≤ 500)?

1

Show question

Question

A fair dice is tossed five times. We build a binomial distribution to X ~ B(5, 0.5) to calculate the probability of getting x heads. What does the cumulative probability P (X ≤ 3) represent?

The probability of getting at most three heads

Show question

Question

A fair dice is tossed 5 times. We build a binomial distribution to X ~ B(5, 0.5) to calculate the probability of getting x heads. What does (1 - P (X ≤ 3)) represent where P (X ≤ 3) is the cumulative probability at x=3?

the probability of getting more than three heads

Show question

Question

State whether the following statement is true or false: the curve for a binomial distribution function is always symmetrical about its peak.

False

Show question

Question

State whether the following statement is true or false: the sum of all probabilities of a binomial distribution can be negative.

False

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Question

What is statistics?

Statistics is the study of probability and data handling. It allows us to make predictions on lots of real life data situations.

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Question

What are the key parts of statistics?

Probability, Distribution Methods and Data Analysis

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Question

What are some topics covered in statistics?

Binomial Distribution, Venn Diagrams, Sampling.

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Question

Give an example of a formula in the formula booklet.

The binomial distribution formula.

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Question

Why is statistics so important?

It is very important as it allows us to analyse and predict based on data from a variety of sources, it can inform key decisions and opinions.

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Question

What is a census?

A census is used to observe every member of the population.

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Question

What is a sample?

A sample is a technique used to observe every member of the population.

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Question

What is a sampling frame?

A sampling frame is the list of all the people or things within the population that can be observed.

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Question

What are some advantages of a census?

A census has the most accurate result as it obverses the whole population.

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Question

What are some advantages of a sample?

A sample is a lot less time consuming than a census, as fewer people have to respond.

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Question

What is quota sampling?

Quota sampling is when the researcher observes sample units and categorises them into groups. Once they have enough sample units per group they will stop any further research.

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Question

What are the advantages of simple random sampling?

There is no bias. Each sampling unit has an equal chance of being selected for research.

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Question

An advantage of opportunity sampling is that it is low cost and easy to do. However, it is unlikely that you will get an accurate representation of the whole population.

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Question

What are the disadvantages of a census?

The disadvantages of a census is that it is time consuming and costly. It is also time consuming to process the large amount of data.

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Question

What is stratified sampling?

This method of sampling involves splitting the population into subgroups based on their behaviour and characteristics, then a random selection is taken from each subgroup.

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Question

What are quantitative data?

Quantitative data are data that involves numbers.

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Question

What are qualitative data?

Qualitative data are data that can be descriptive.

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