Statistics FORMULAS.

Statistics is the field of Maths dealing with the collection, analysis, and presentation of data. Statistics is a theory-heavy concept. Students should familiarise themselves with the language and concepts surrounding statistics by using the textbook. These concepts include: data, numerical data, categorical data, discrete data, continuous data, ordinal data, univariate data, bivariate data, primary data, secondary data, surveys, experiments, biasness, population, random sample, mean, mode, median, range; interquartile range, variability, frequency distributions and tables, grouped frequency distributions, standard deviation, empirical rule, inferential statistics, confidence intervals, sample proportion, margin of error and hypothesis testing.

List of charts that students should know

Bar Charts, Scatter Plots, Line Plots, Pie Charts, Histograms, Positive Skew, Negative Skew, Normal Distribution, Back-to-back Stem and Leaf Diagrams, Correlation, Positive Correlation, No Correlation, Negative Correlation, Causal Relationship.

FORMULAS AND FACTS. FOR STATISTICS.

1. Mode: This is the most common number in the list. 2. Median: When numbers are placed in ascending/descending order the median is the middle number.

3. Mean: This is the average of the list. To find the average, students must add up all the numbers and then divide by the number of numbers. The symbol for mean is μ. 4. Range: The highest value in a set of data minus the lowest value. This shows the spread of the data. 5. Lower quartile: When data is arranged in descending/ascending order this is the value one quarter of the way into the data. The lower quartile is written as Q1. 6. Upper quartile: When data is arranged in descending/ascending order this is the value three quarters of the way into the data. The upper quartile is written as Q3. 7. Interquartile range: The upper quartile minus the lower quartile. It can be found by Q3–Q1. 8. Standard deviation: This is a measure of the spread σ. of the values about the mean. The symbol is The formulae are on page 33 in the mathematical tables. l Mean from a list of n numbers: l Mean from frequency table: l

Standard deviation from a list of n numbers: l Standard deviation from frequency table:

Empirical rule: For many large populations, the empirical rule provides an estimate of the approximate percentage of observations that are contained within one, two or three standard deviations of the mean.

∓ 1σ lμ covers

68% of the sample

∓ σ lμ 2 covers

95% of the sample

∓ σ lμ 3 covers almost all the sample

(99.7%). What is inferential statistics? Inferential statistics is the branch of statistics that uses probability and statistics to draw conclusions from data that are affected by random variation. In this section, students should be able to:

1. Estimate the value of a population proportion.

2. Calculate the margin of error for a sample. 3. Construct a confidence interval. 4. Apply the empirical rule to confidence intervals. 5. Test a hypothesis about a population proportion.

2017 PAPER 2 – QUESTION 9(C).

In the days following the UK referendum, a survey was conducted in Ireland on attitudes towards a British exit from the EU. (i) 1,200 people were surveyed. Find the margin of error of this survey. Write your answer as a percentage. Give your answer correct to the nearest percent. The formula for the margin of error is: where n is the size of the sample. (ii) In the Irish survey 578 of the 1,200 surveyed agreed that a UK exit would have a negative effect on the Irish economy. Use your answer to part c(i) above to create a 95% confidence interval for the proportion of the Irish population who agreed that a UK exit would have a negative effect on the Irish economy.

A confidence interval for the true proportion at the 95% confidence interval is given by where n is the sample size, p is the population proportion and p̂ is the sample proportion. We can say that the 95% confidence interval lies between inside this interval. (iii) After the survey, a political party claimed that 53% of the Irish population agreed that the decision of the UK to leave the EU would have a negative effect on the Irish economy. Use your answer to part c(ii) above to conduct a hypothesis test, at the 5% level of significance, to test the party’s claim. Give your conclusion on the context of the question.

Null hypothesis: = p =53%

H0 Alternative hypothesis: = p ≠ 53% H1

53% is not within the confidence interval, as found in part (c)(ii).

45.17 < p <51.17

Reject the null hypothesis and accept the

H0 alternative hypothesis H1. To conclude our answer, we can now state that the political party’s claim is false.