6th Year Maths Statistics 2 Past Paper 2024/2025 PDF

6^th^ Year Maths Statistics 2 http://www.turnerandtownsend.com/1111/Rathgar-Dublin\_2110\_245EG0.jpg.img 2024/2025 Ms Thorp **Name: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\ ** **1. Hypothesis Testing** A [hypothesis] is a statement about a statistic which has yet to be prov...

6^th^ Year Maths Statistics 2 http://www.turnerandtownsend.com/1111/Rathgar-Dublin\_2110\_245EG0.jpg.img 2024/2025 Ms Thorp **Name: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\ ** **1. Hypothesis Testing** A [hypothesis] is a statement about a statistic which has yet to be proven/disproven. A [hypothesis test] is a way of proving the statement true/false. The statement being tested is called the [null hypothesis] or H~0.~ It is usually a statement of no effect or no difference. We tend to start with an attitude of scepticism. After carrying out our investigation we may prove the null hypothesis is not true. We then reject the null hypothesis. If we have not proved it incorrect, we fail to reject the null hypothesis. It may still be incorrect, but we have not managed to prove that yet! Example 1: Is the new drug effective in treating COVID-19? H~0~ = The new drug is not effective H~A~ = the alternative hypothesis = the new drug is effective We investigate and then either reject H~0~ OR fail to reject H~0~ **2. Population proportion p and sample proportion** [\$\\widehat{\\mathbf{p}}\$]{.math.inline} Population = entire group of interest. We carry out a census on them. Sample = selection of the population. We carry out a sample survey on them. Caution needs to be taken in choosing n, the number of people in the sample - not too many (expensive and time consuming) - not too few (larger percentage error, results unreliable) - who is picked (not a biassed sample) The population proportion P is a fixed number. We will want to find it. For example what is the percentage of people in Ireland that are left-handed. The sample proportion [*p̂*]{.math.inline} is the result from the sample survey. For example 60 out of 400 people surveyed were left-handed = 15% We could get an entirely different value if I surveyed a different 400 people. It depends on chance. It has a margin of error associated with it. We might infer that the true amount of left-handed people in the population is 15% ± 5%. 3. **Margin of Error** i. 1.25% ii. 3% ~\ ~ i. margin of error ii. the confidence interval iii. four weeks later a survey of 1200 people is carried out to check if there has been a change in support for the election. 696 are in favour of the election. State the null hypothesis and whether we reject or fail to reject it. ~\ ~ 4. **Confidence Interval** \ [*Sample* *Proportion* − *M*. *of* *Error*  ≤ *Population* *Proportion* ≤ *Sample* *Proportion* + *M*. *of* *Error*]{.math.display}\ or [*p̂* − *E*  ≤ *P* ≤ *p̂* + *E*]{.math.inline} or [\$\\widehat{p} - \\frac{1}{\\sqrt{n}}\\ \\leq P\\ \\leq \\ \\widehat{p} + \\frac{1}{\\sqrt{n}}\$]{.math.inline} (When we say at a 95% level of confidence we mean that if we carried out a lot of surveys, 95% of the confidence intervals formed would contain the true population proportion.) ------------ ------------------------ ------------------------------------------ Population Sample Mean \ \ [*μ*]{.math.display}\ [\$\$\\overline{x}\$\$]{.math.display}\ Proportion p [*p̂*]{.math.inline} ------------ ------------------------ ------------------------------------------ Notation: 5. **Standard error for the sample proportion,** [**σ**]{.math.inline} *We will now use a more accurate formula for margin of error.* 6. **Confidence Intervals** \ [*p̂* − 1.96σ   ≤ *P* ≤ *p̂* + 1.96σ ]{.math.display}\ *or* *Note:* *If p (or* [*p̂*)]{.math.inline} is almost equal to [\$\\frac{1}{2}\$]{.math.inline}, Then Margin of Error = 1.96[\$\\sqrt{\\frac{\\frac{1}{2}(1 - \\frac{1}{2})}{n}}\$]{.math.inline} *= 1.96*[\$\\sqrt{\\frac{1}{4n}}\$]{.math.inline} [\$\\approx \\frac{1}{\\sqrt{n}}\$]{.math.inline} *This is where the approximation for Margin of error as* [\$\\frac{1}{\\sqrt{n}}\$]{.math.inline}*, that we initially used in our questions, came from. But only use this if we don't have enough info to use anything more accurate, or if* [\$\\widehat{p} \\approx \\frac{1}{2}\$]{.math.inline}, or it told to! Example 4: A poll of 2020 people is carried out. 404 people said they take a duvet day at least once a year. Form a 95% confidence interval for the proportion of people that take a duvet day. Example 5: In a by-election of 900 people, the results suggest 35% will vote for candidate A. i. Find the standard error ii. *If a sample of 400 people was taken instead would the standard error increase or decrease. Explain your answer.* iii. *Candidate A wants to find his level of support to within ±*[\$\\frac{1}{2}\$]{.math.inline}*%. What sample size is required?* 7. ***Hypothesis Test*** 1. H~0~ = H~A~ = 2. [*p̂*]{.math.inline} = 3. *95% Margin of Error = (* [1.96*σ*]{.math.inline} *or* [\$1.96\\sqrt{\\frac{P(1 - P)}{n}}\$]{.math.inline}*)* 4. *95% Confidence Interval = (* [*p̂* − 1.96σ   ≤ *P* ≤ *p̂* + 1.96σ ]{.math.inline}*)* 5. *Show Confidence Interval on a diagram* 6. *If population proportion is within the CI:* 7. *State the conclusion in words.* 8. **Distributions of sample means & Central Limit Theorem (CLT)** - The population has a mean [μ ]{.math.inline}and a standard deviation [*σ*]{.math.inline} - If we take a large number of samples, each of size n - Each sample has a mean, [\$\\overline{x}\$]{.math.inline} - Each sample has a standard deviation, s - Set of sample means = - If you represent the set of sample means with a curve, the distribution will have 3 interesting properties. We call these properties the Central Limits Theorem. 9. **The Central Limit Theorem** 1. Distribution of sample means = Normal 2. mean of sample means = mean of population 3. standard deviation of sample means = [\$\\frac{\\text{Standard\\ deviation\\ of\\ population}}{\\sqrt{n}}\$]{.math.inline} The central limit theorem works for large samples only (i.e. n \> 30) 10. **Standard Error** \ [\$\$\\sigma\\ = \\ \\frac{\\sigma}{\\sqrt{n}}\$\$]{.math.display}\ - If Population Proportions: *(* [*p̂* − 1.96σ   ≤ *P* ≤ *p̂* + 1.96σ ]{.math.inline}*)* - If Population Means: *(* [\$\\overline{x} - 1.96\\text{σ\\ }\\ \\leq \\mu\\ \\leq \\ \\overline{x} + 1.96\\text{σ\\ }\$]{.math.inline}*)* 11. **P-Values** To find the Z-Score (often called the Test Statistic, T) to help you work out the p-values: \ [\$\$T = Z = \\ \\frac{\\overline{x} - \\ \\mu}{\\frac{\\sigma}{\\sqrt{n}}}\$\$]{.math.display}\ Note: Can use p-values to decide on H~0~. OR If they don't specifically ask you to find the p-value, you can just get the z-score (also called Test Statistic) and compare it to the critical values of [ ± 1.96]{.math.inline} If T lies outside the interval -- Reject H~0~ If T lies inside the interval -- Fail to Reject H~0~