Ho me Inferential Statistics 2015-2016 Ho me Overview L.C. O.L.

L.C. H.L. 1. Sampling variability 2. Confidence intervals for population proportion. 3. Hypothesis testing using confidence intervals 1. Normal distribution 2. Sampling variability

3. Distribution of sample means 4. Confidence interval for population mean 5. Distribution of sample proportions 6. Confidence interval for population proportion 7. The margin of error 8. Hypothesis testing using pvalues Ho

me Inferential Statistics Leaving Cert. Ordinary Level Ho me Statistical Inference Ask a question Census

Gather data Sample Analyse data Can we reliably use the results from a single sample to make conclusions about a population? Draw conclusions Ho me Inferences about

proportions Ho me Sampling variability What proportion of students keep their mobile phone under their pillow at night? Ho me

Sampling variability Make your own statement The proportion of all students who keep their mobile phone under their pillow at night is _____Sample Statement 1 2 3 4 5 6

7 Ho me Sampling variability Different samples will yield different results (due to random nature of sampling) How then can we use a single sample to draw conclusions about a population?

Ho me Sampling variability Ho me Sampling variability How can we capture this uncertainty in our statement?

The proportion of all students who keep their mobile phone under their pillow at night is _____ The proportion of students who keep their mobile phone under their pillow at night is between 0.1 and 0.45? Ho me A confidence interval How do we decide on this range if we

have one sample only? How confident can we be that this range captures the population proportion? Ho me The 95% confidence interval I can be 95% confident that the population proportion lies between &

Ho me What does 95% confidence mean? Ho me The margin of error is often referred to as the margin of error?

Why do you think this is? Ho me The margin of error How could you make your margin of error small? Why should depend on n? n

1 1 10 0.316 100

0.1 500 0.045 1000 0.032 2000

0.022 5000 0.014 n Ho me The margin of error

Why might you want to make your margin of error small? Ho me The power of statistics 4.6 million 1.36 billion

Assessment of student understanding Ho me Assessment of student understanding Ho me

Ho me A political party claims that it has the support of 24% of the electorate. In a sample of 1111 voters, 243 state that they support the party. Is there sufficient evidence to reject the partys claim, at the 5% significance level? Step 1. State the null & alternative hypotheses 0.24 of the electorate support the party

The proportion of the electorate that support the party is not 0.24 0.24 of the electorate do not support the party intermediate

step to help students : p0.24 : p0.24 with language Null Hypothesis Hypothesis tests

Ho me Hypothesis tests Step 2. Build a 95% confidence interval for the population proportion. I am 95% confident that the population proportion lies between and . i.e. the 95% confidence interval for the population proportion is: Step 3. Make conclusion based on whether hypothesised proportion is inside or outside the

Since is within the 95% confidence interval I confidence interval. fail to reject the null hypothesis. There is insufficient evidence to reject the partys claim at the 5% significance level. Ho me Hypothesis tests

Ho me Inferential Statistics Leaving Cert. Higher Level Ho me Statistical Inference

Ask a question Census Gather data Sample Analyse data Can we reliably use the results from a single sample to make conclusions about a population? Draw conclusions Ho me

Prior knowledge Ho me The Normal Distribution de Moivre Gauss

Laplace = Qutelet Ho me The Normal Distribution

The heights of Irish males is normally distributed with a mean of 176 cm and a standard deviation of 6.5 cm. 1. What proportion of Irish males have heights between 169.5 cm and 182.5 cm? 2. What proportion of Irish males have heights between 166.5 cm and 185.5 cm? 3. What proportion of Irish males have heights greater than 190 cm? 4. What proportion of Irish males have heights equal to 190 cm? 5. If I choose an Irish male at random, what is the probability he will have a height greater than 190 cm?

6. If I choose an Irish male at random, what is the probability he will have a height of 190 cm? Ho me Sampling variability A good understanding of sampling variability lays the foundations for confidence intervals hypothesis testing Sketching the distribution of the

sample statistic is a key skill students should develop Ho me Inferences about means Ho me Sampling variability

What is the mean schoolbag weight for Irish secondary-school students? 9 . 4 5. 3 2 . 5

4.5 Ho me Sampling variability Make your own statement The mean schoolbag weight for all Irish postprimary students is _____ Sample 1 2 3

4 5 6 7 Statement Ho me Sampling variability Sampling variability means we

cannot equate the results from a single sample with those for a population. Ho me The distribution of sample means In spite of this we can still use a single

sample to make a valid statement about a population To do so we need to understand all the possible means we can get when we choose a sample. Ho me The distribution of sample means Different samples give different means but the distribution of sample means is normal (for large

sample sizes). Ho me The distribution of sample means The centre of the distribution () is identical to the population centre (). Ho me

The distribution of sample means The distribution is more compact than the population. Ho me The distribution of sample means Why is the distribution of sample means more compact than the population?

Ho me The distribution of sample means How does the spread of the distribution of sample means compare to the population? . = .

Ho me The 95% confidence interval This means I can say with 95% confidence my value lies within 1.96 standard deviations of the centre of my distribution(). This means I can also say with 95% confidence that the centre of my

distribution() lies within 1.96 standard deviations of my value. Ho me The 95% confidence interval I can say with 95% confidence that lies within of . I can say with 95% confidence that lies within of . I can say with 95% confidence that

lies within of . Constructing a 95% confidence interval Use for building confidence interval if you know its value. Otherwise use s. 2.57 64

I can say with 95% confidence that 4.6 lies within of I can say with 95% confidence that lies 2.57 2.57 4.6 1.96 4.6+ 1.96

within of 4.6 64 64 Ho me Constructing a 95% confidence interval 2.57 2.55 64 64

2.55 2.55 4.6 1.96 4.6+1.96 64 64 Ho me

Assessment of student understanding Ho me Assessment of student understanding Ho me

Ho me Summary Due to sampling variability I cannot say . Due to the normal shape of the distribution of sample means I can say with 95% confidence that lies within of .

1.96 + 1.96 Ho me Inferences about proportions

Ho me Sampling variability What proportion of students keep their mobile phone under their pillow at night? Ho me Sampling variability

Make your own statement The proportion of all students who keep their mobile phone under their pillow at night is _____Sample Statement 1 2 3 4 5 6 7

Ho me Sampling variability Sampling variability means we cannot equate the results from a single sample with those for a population. ^

The distribution of sample proportions In spite of this we can still use a single sample to make a valid statement about a population To do so we need to understand all the possible proportions we can get when we choose a sample. Ho me

The distribution of sample proportions Different samples give different proportions but they all follow a normal distribution. Ho me The distribution of sample proportions The centre of the distribution () is

identical to the population proportion (). Ho me The distribution of sample proportions The standard deviation of the distribution is given by Ho

me Ho me The 95% confidence interval This means I can say with 95% confidence that my value lies within standard deviations of the centre of the distribution. This also means I can say with 95% confidence that the centre of the

distribution lies within standard deviations of my value. Ho me The 95% confidence interval I can say with 95% confidence that lies within of . I can say with 95% confidence that lies within of . I can say with 95% confidence that

lies within of . Constructing a 95% confidence interval Use for building confidence interval if you know its value. Otherwise use .

Ho me (0.2)(0.8) 30 ^ I can say with 95% confidence that 0.2 lies within of . I can say with 95% confidence that lies within of 0.2.

Constructing a 95% confidence interval The 95% confidence interval for the population proportion is: ( 0.2 )( 0.8 ) (0.2)(0.8) 0.21.96 0.2+1.96 30 30

Ho me Ho me The margin of error

( )( 1 ) 1 1.96 Assessment of student understanding Ho

me Ho me o o Means =

= I can say with 95% confidence that lies within of my value. Proportions ( ) ( ) ^ =

^= I can say with 95% confidence that lies within of my value. Formulae Summary

Ho me Hypothesis testing using p-values Hypothesis tests are based on understanding the properties of the distribution of the sample means (or sample proportions). Ho me

Hypothesis testing using p-values The mean amount of time spent daily on homework & study by Leaving Cert. students in 2013-2014 was 5.4 hours with a standard deviation of 1.8 hours. A guidance counsellor surveys 50 Leaving Cert. students in his school during 2014-2015 and finds that the mean amount of time spent on homework is 5.1 hours. By carrying out a hypothesis test at the 5% significance level, determine if the results for 2014-2015 are consistent with those for 20132014.

Ho me Hypothesis testing using p-values H0: The mean amount of time spent studying by Leaving Cert. students in 2014-2015 is 5.4 hours. H1: The mean amount of time spent studying by Leaving Cert. students in 2014-2015 is not 5.4 hours.

Ho me The language of a hypothesis test The term null hypothesis comes from the idea of a null effect or no change so should be stated as such i.e. as a statement of no change H0: The mean amount of time spent studying by Leaving Cert. students in 2014-2015 is 5.4 hours The mean amount of time spent daily on homework & study by Leaving Cert. students in 2013-2014 was 5.4 hours with a

standard deviation of 1.8 hours. A guidance counsellor surveys 50 Leaving Cert. students in his school during 2014-2015 and finds that the mean amount of time spent on homework is 5.1 hours. Ho me Hypothesis testing using p-values If H0 is true wed expect the distribution of sample means to be: 1.8

= 50 =5.4 Ho me Hypothesis testing using p-values If H0 is true how likely am I to get a sample mean of 5.1 due to variability?

Because of the hypothesis I need to determine how likely I am to get a sample mean of 5.1 or 5.7? Because of the properties of the normal distribution I need to determine how likely I am to get a sample mean of less than (or equal to) 5.1 or greater than (or equal to) 5.7. Ho me Hypothesis testing using p-values Use z-scores to determine this probability

5.4 5.1 ||= =1.18 1.8 50 (|| 1.18 )=2(1 0.8810) 0.238 Ho me Hypothesis testing using p-values

If this probability is really small, this implies that the sampling distribution is unlikely to be centred on the hypothesised value (assuming the given standard deviation) How small is small? 5% 0.05 Ho me

What does the p-value mean? probability of getting results at least as unusual as the observed mean given that the null hypothesis is true. What does the level of significance mean? 1. The probability boundary around which you either reject or fail to reject the null hypothesis. If p significance level, fail to reject the null hypothesis. If p significance level, reject the null

hypothesis. 2. The probability of rejecting the null hypothesis even if its true. Ho me Ho me The syllabus

Assessing student understanding Ho me The syllabus Assessing student understanding The syllabus Alternative approach using p

values Ho me Ho me Use if known. Otherwise use . The syllabus

Alternative approach using p values Ho me Assessing student understanding Use if known. Otherwise use s. Ho me

Assessing student understanding Ho me Assessing student understanding Ho me Assessing student understanding

Ho me Summary Hypothesis testing is built on a good understanding of z-scores the distribution of the sample statistic