Inferential Statistics – Outcomes Make decisions based on the empirical rule. Recognise the concept of a hypothesis test. Calculate the margin of error for a population proportion. Construct a 95% confidence interval. Conduct a hypothesis test on a population proportion using the margin of error.

Use Empirical Rule Recall the normal distribution. It describes data sets where: Most of the data is close to the mean. There is a trend towards less data as you get farther from the mean. e.g. most people are close to the mean height whereas few people are very tall or very short.

Use Empirical Rule For data that is perfectly normally distributed, we can describe how much data is a certain distance from the mean: You need to remember these numbers

Use Empirical Rule e.g. A group of students are each asked to solve a particular problem. The time taken for each student to solve the problem is recorded. The mean time taken (𝜇) is 20 minutes and the standard deviation (𝜎) is 2.5 minutes. Complete the following sentence: “95% of students took between _____ and _____ minutes to solve the problem. Make one other statement about the times taken by students to solve the problem using the empirical rule. 68% 𝜇−𝜎, 𝜇+𝜎 95% 𝜇−2𝜎, 𝜇+2𝜎 99.7% 𝜇−3𝜎, 𝜇+3𝜎

Use Empirical Rule For each of the following, find the intervals in which 68%, 95% and 99.7% of the population lies, given that the mean (𝜇) and standard deviation (𝜎) are from a normal distribution. 𝜇=40 𝜎=2 𝜇=110 𝜎=5 𝜇=62 𝜎=3.5 𝜇=125 𝜎=8 𝜇=49 𝜎=4.5 𝜇=15 𝜎=1.4

Use Empirical Rule If 95% of Irish women are between 148.3 cm and 178.7 cm, find the mean and standard deviation of the heights using the empirical rule. If 68% of a population has an IQ between 85 and 115, find the mean and standard deviation of the IQs using the empirical rule. If 84% of adorable, fluffy kittens weigh 1.8 kg or less with a mean of 1.6 kg, find the standard deviation of adorable, fluffy kitten weights.

Use Empirical Rule 2012(S) P2 Q6 The heights in 2011 of Irish males born in 1992 are normally distributed with mean 178·8 cm and standard deviation 7·9 cm. Use the empirical rule to complete the following sentence: “95% of nineteen-year-old Irish men are between _____ and _____ in height”. Use the empirical rule to make one other statement about the heights of nineteen-year-old Irish men.

Use Empirical Rule 2015 P2 Q9 The mean height of the boys in the sample is 166·7 cm and the standard deviation of their height is 8·9 cm. Assuming that boys’ heights are normally distributed, use the Empirical Rule to find an interval that will contain the heights of approximately 95% of all boys.

Recognise the Concept of Hypothesis Test e.g. Grace is a vet who wants to test her new treatment for myxomatosis. The current standard treatment has a 50% effectiveness rate. Grace tries out her new treatment on a rabbit and cures it! Does her new treatment have a 100% effectiveness rate?

Recognise the Concept of a Hypothesis Test e.g. Daniel wants to check if his coin is biased. An unbiased coin will have a 50% chance of turning up heads. He flips the coin ten times and it turns up heads seven of those times. Is the coin biased?

Recognise the Concept of a Hypothesis Test e.g. Robyn is doing a survey to find out if people’s attitudes to reducing the minimum age of candidacy for President to 21 has changed since the referendum in 2015. 73.06% of people voted against the referendum in 2015. According to Robyn’s survey, 68% of people would currently vote against the same amendment. Have people’s attitudes changed?

Recognise the Concept of a Hypothesis Test A hypothesis test consists of: Writing down the null hypothesis – the idea that a difference between two results is only due to chance. e.g. Grace’s treatment is actually 50% effective, like the existing treatment and her test showed 100% effective by chance. e.g. Daniel’s coin has a 50% heads rate and turned up seven heads instead of five by chance. e.g. People still have a 73.06% reject proportion and Robyn’s sample was somehow biased by chance. The null hypothesis is the sceptical position.

Recognise the Concept of a Hypothesis Test A hypothesis test consists of: Writing down the alternative hypothesis – the idea that there is a real difference between two results. e.g. Robyn’s treatment is not 50% effective. e.g. Daniel’s coin does not have a 50% rate of turning up heads. e.g. The proportion of people who would reject the referendum is not 73.06%.

Recognise the Concept of a Hypothesis Test A hypothesis test consists of: Finding the margin of error in the experiment. Constructing a confidence interval for the experiment. Deciding whether the null hypothesis is compatible with the confidence interval that has been constructed. Rejecting or failing to reject the null hypothesis.

Recognise the Concept of a Hypothesis Test e.g. write down a suitable null hypothesis and alternative hypothesis for each of the following situations: The ISPCA stated that 30% of households keep a dog. A survey of 400 students found that 112 of their households kept a dog. A survey of 1000 voters suggested that 350 would vote for the No Homework Party in the next election. The leader of the No Homework party states that the true proportion is 40%. A library stated that 12% of returned books were overdue. A random sample of 200 returned books revealed that 15 were overdue.

Calculate the Margin of Error The margin of error for a sample is given by: Formula: 𝑚𝑎𝑟𝑔𝑖𝑛 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟= 1 𝑛 , where 𝑛 is the size of the sample. e.g. calculate the margin of error for the following sample sizes as a percentage to two decimal places: 𝑛=1 𝑛=10 𝑛=100 𝑛=250 𝑛=1000 This formula is not in the F&T book

Calculate the Margin of Error e.g. Given the following margins of error, calculate the sample size. 0.4472 25.82% 14.14% 0.7071 15.81% 12.91% 0.2236 margins of error should be in decimal form when calculating sample size

Construct a 95% Confidence Interval With any experimental result on a sample, we will have uncertainty about how well it describes the population. A confidence interval represents the region around our result where we think the true result should be. We construct a confidence interval for our sample proportion ( 𝑝 ) by setting upper and lower bounds with the margin of error: Formula: 𝑝 − 1 𝑛 , 𝑝 + 1 𝑛

Construct a 95% Confidence Interval 𝑝 − 1 𝑛 , 𝑝 + 1 𝑛 e.g. Daniel wants to check if his coin is biased. An unbiased coin will have a 50% chance of turning up heads. He flips the coin ten times and it turns up heads seven of those times. Construct a 95% confidence interval for this experiment.

Construct a 95% Confidence Interval 𝑝 − 1 𝑛 , 𝑝 + 1 𝑛 e.g. Robyn is doing a survey of 500 people to find out their attitudes to reducing the minimum age of candidacy for President to 21 has changed since the referendum in 2015. 73.06% of people voted against the referendum in 2015. According to Robyn’s survey, 68% of people would currently vote against the same amendment. Construct a 95% confidence interval for this experiment.

Construct a 95% Confidence Interval 𝑝 − 1 𝑛 , 𝑝 + 1 𝑛 e.g. Grace is a vet who wants to test her new treatment for myxomatosis. The current standard treatment has a 50% effectiveness rate. Grace tries out her new treatment on a rabbit and cures it! Construct a 95% confidence interval for this experiment.

Conduct a Hypothesis Test If the population proportion (𝑝) lies outside the confidence interval, we reject the null hypothesis. If the population proportion lies inside the confidence interval, we fail to reject the null hypothesis. e.g. We fail to reject Daniel’s null hypothesis because 50% lies inside his confidence interval. e.g. We reject Robyn’s null hypothesis because 73.06% lies outside her confidence interval. e.g. We fail to reject Grace’s null hypothesis because 50% lies inside her confidence interval.

Conduct a Hypothesis Test 5% level of significance corresponds to a 95% confidence interval State the null hypothesis and alternative hypothesis, and state your conclusion in each case below at the 5% level of significance: The ISPCA stated that 30% of households keep a dog. A survey of 400 students found that 112 of their households kept a dog. A survey of 1000 voters suggested that 350 would vote for the No Homework Party in the next election. The leader of the No Homework party states that the true proportion is 40%. A library stated that 12% of returned books were overdue. A random sample of 200 returned books revealed that 15 were overdue.

Conduct a Hypothesis Test 2015(S) OL Q1 A survey is being conducted of voters’ opinions on several different issues. What is the overall margin of error of the survey, at 95% confidence, if it is based on a simple random sample of 1111 voters? A political party had claimed that it has the support of 24% of the electorate. Of the voters in the sample above, 243 stated that they support the party. Is this sufficient evidence to reject the party’s claim, at the 5% level of significance?

Conduct a Hypothesis Test 2015(S) OL Q2 A widget-manufacturing company repeatedly asserts that 80% of traders recommend their brand of widget. In a survey of 40 traders, 24 said that they would recommend the company’s widget. Use a hypothesis test at the 5% level of significance to decide whether there is sufficient evidence to reject the company’s claim. State clearly the null hypothesis and your conclusion.

Conduct a Hypothesis Test 2013 HL P2 Q7 The company repeatedly asserts that 70% of their customers are satisfied with their overall service. Use a hypothesis test at the 5% level of significance to decide whether there is sufficient evidence to conclude that their claim is valid in May. Write the null hypothesis and state your conclusion clearly. A manager of the airline says: “If we survey 2000 passengers from June on, we will halve the margin of error in our surveys.” Is the manager correct?