Our statistics KS5 helps student to employ the statistical enquiry cycle to help make sense of data trends and to solve statistical problems in a variety of contexts, such as psychology, biology, geography, business and the social sciences.

# Statistics KS5

£75.00

### Description

**Statistics KS5**

## Objectives and aims:

Our Statistics KS5 solution meets the needs to develop understanding, skills and knowledge required for students at KS5 level. We incorporate fun and intriguing technologies to teach some of the hardest statistics concepts. Our solution emphasises how mathematical ideas are interconnected and how statistics can be applied to model situations statistically using probability, permutations and combinations. It prepares students for further study and employment in a wide range of disciplines involving the use of statistics. It helps student to employ the statistical enquiry cycle to help make sense of data trends and to solve statistical problems in a variety of contexts, such as psychology, biology, geography, business and the social sciences.

**Statistics KS5 Contents:**

### Numerical Measures, graphs and diagrams:

- interpret statistical diagrams including bar charts, stem and leaf diagrams, box and whisker plots, cumulative frequency diagrams, histograms (with either equal or unequal class intervals), time series and scatter diagrams
- know the features needed to ensure an appropriate representation of data using the above diagrams, and how misrepresentation may occur
- justify appropriate graphical representation and comment on those published.
- compare different data sets, using appropriate diagrams or calculated measures of central tendency and spread: mean, median, mode, range, interquartile range, percentiles, variance and standard deviation
- calculate measures using calculators and manual calculation as appropriate
- identify outliers by inspection and using appropriate calculations
- determine the nature of outliers in reference to the population and original data collection process
- appreciate that data can be misrepresented when used out of context or through misleading visualisation

### Probability:

- know and use language and symbols associated with set theory in the context of

probability - represent and interpret probabilities using tree diagrams, Venn diagrams and twoway tables
- calculate and compare probabilities: single, independent, mutually exclusive and

conditional probabilities - use and apply the laws of probability to include conditional probability
- determine if two events are statistically independent

### Population and samples:

- Understand and use the equation of a straight line.
- know both simple (without replacement) and unrestricted (with replacement) random samples
- know how to obtain a random sample using random numbers tables or random numbers generated on a calculator
- evaluate the practical application of random and non-random sampling techniques: simple random, systematic, cluster, judgemental and snowball, including the use of stratification (in proportional and disproportional ratios) prior to sampling taking place
- know the advantages and limitations of sampling methods
- make reasoned choices with reference to the context in which the sampling is to take place, examples include, but are not limited to: market research, exit polls, experiments and quality assurance
- understand the practical constraints of collecting unbiased data

### Introduction to probability distributions

- know and use terms for variability: random, discrete, continuous, dependent and independent
- calculate probabilities and determine expected values, variances and standard deviations for discrete distributions
- use discrete random variables to model real-world situations
- know the properties of a continuous distribution
- interpret graphical representations or tabulated probabilities of characteristic discrete random variables
- interpret rectilinear graphical representations of continuous distributions

### Binomial Distribution:

- know when a binomial model is appropriate (in real world situations including modelling assumptions)
- know methods to evaluate or read probabilities using formula and tables
- calculate and interpret the mean and variance

### Normal Distributions:

- know the specific properties of the normal distribution, and know that data from such an underlying population would approximate to having these properties, with different samples showing variation
- apply knowledge that approximately 2/3 of observations lie within µ ± σ , and equivalent results for 2σ and 3σ
- determine probabilities and unknown parameters with a normal distribution
- apply the normal distribution to model real world situations
- use the fact that the distribution of X has a normal distribution if X has a normal distribution
- use the fact that the normal distribution can be used to approximate a binominal distribution under particular circumstances

### Correlation and Linear Regression:

- calculate (only using appropriate technology – calculator) and interpret association using Spearman’s rank correlation coefficient or Pearson’s product moment correlation coefficient
- use tables to test for significance of a correlation coefficient
- know the appropriate conditions for the use of each of these methods of calculating correlation and determine an appropriate approach to assessing correlation in context
- calculate (only using appropriate technology – calculator) and interpret the coefficients for a least squares regression line in context; interpolation and extrapolation, and use of residuals to evaluate the model and identify outliers

### Introduction to hypothesis testing:

- use and demonstrate understanding of the terms parameter, statistic, unbiased and standard error
- know and use the language of statistical hypothesis testing: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, and acceptance region and p-value.
- know that a sample is being used to make an inference about the population and appreciate the need for a random sample and of the necessary conditions
- choose the appropriate hypothesis test to carry out in particular circumstances
- conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context using exact probabilities or, where appropriate, a normal approximation
- conduct a statistical hypothesis test for the mean of a normal distribution with known or assumed, from a large sample, variance and interpret the results in context
- know the importance of appropriate sampling when using hypothesis tests and be able to critique the conclusions drawn from rejecting or failing to reject a null hypothesis by considering the test performed

### Contingency Tables:

- construct contingency tables from real data, combining data where appropriate, and interpret results in context
- use a χ2 test with the appropriate number of degrees of freedom to test for independence in a contingency table and interpret the results of such a test
- know that expected frequencies must be greater than, or equal to, 5 for a χ2 test to be carried out and understand the requirement for combining classes if that is not the case

### One and two sample non-parametric tests

- use sign or Wilcoxon signed-rank tests to investigate population median in single sample tests and also to investigate for differences using a paired model
- use the Wilcoxon rank-sum test to investigate for difference between independent samples

### Bayes’ Theorem

- calculate and use conditional probabilities to include Bayes’ theorem for up to three events, including the use of tree diagrams.

### Probability Distributions

- know the use and validity of distributions which could be appropriate in a particular real-world situation: binomial, normal, Poisson and exponential
- evaluate the mean and variance of linear combinations of independent random variables.

### Experimental Design

- know and discuss issues involved in experimental design: experimental error, randomisation, replication, control and experimental groups, and blind and double blind trials
- know the benefits of use of paired comparisons and blocking to reduce experimental error
- use completely random and randomised block designs

### Sampling, estimates and resampling

- Use and demonstrate understanding of terms parameter, statistic, unbiased and standard error
- know the use of the central limit theorem in the distribution of X where the initial distribution, X , is not normally distributed and the sample is large

### Hypothesis testing, significance testing, confidence intervals and power

- Use confidence intervals for the mean using z or t as appropriate, interpreting results in practical contexts
- know that a change in sample size will affect the width of a confidence interval
- evaluate the strength of conclusions and misreporting of findings from hypothesis tests, including the calculation and importance of the power of a hypothesis test
- know that sample size can be changed to potentially elicit appropriate evidence in a hypothesis test
- interpret type I and type II errors, in hypothesis testing and know their practical meaning
- calculate the risk of a type II error
- know the difference and advantages of using critical regions or p-values as appropriate in real-life contexts in all tests in these subject content

### Hypothesis testing for 1 and 2 samples

- know how to apply knowledge about carrying out hypothesis testing to conduct

tests for the:- mean of a normal distribution with unknown variance using the t distribution
- difference of two means for two independent normal distributions with known variances
- difference of two means for two independent normal distributions with unknown but equal variances
- difference between two binomial proportion

- interpret results for these tests in context

### Paired Tests

- use sign, Wilcoxon signed-rank or paired t-test, understanding appropriate test selection and interpreting the results in context

### Exonential and Poisson distributions

- determine when a Poisson model is appropriate (in real world situations including modelling assumptions)
- determine when an exponential distribution is appropriate (and its relationship to the Poisson distribution as a model of the times between randomly occurring Poisson events)
- evaluate probabilities for Poisson and exponential distributions and know the corresponding mean and variance

### Goodness of fit

- conduct a statistical goodness of fit test for binomial, Poisson, normal and Exponential distributions or for a specified discrete distribution using ∑ (O – E) 2/E as an approximate χ 2 statistic

### Analysis of variance

- conduct one-way analysis of variance, using a completely randomised design with appreciation of the underlying model with additive effects and experimental errors distributed as N(0, σ2 )
- conduct two-way analysis of variance without replicates, using a randomised block design with blocking
- identify assumptions and interpretations in context

### Effect size

- know the notion of effect size as a complementary methodology to standard significance testing, and apply in authentic contexts
- know and use Cohen’s d in simple situations