STAT170 Workshop Notes prepared by Nan Carter for Numeracy ...

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The summary statistics are: Group number x-bar SD Activity group 26 51.48 11.01 Control group 28 41.52 17.15 Is there a difference, on average, between the two groups of students? State any assumptions that you have made. ANSWERS: Q1: pooled SD = 5.2679 SE(y1bar – y2bar) = 0.6164 t= 0.649 df = 293 p-value < 0.05 Q2: pooled sd = 5.5873 SE(y1bar – y2bar) = 2.2512 t= 2.221 0.02
Week 9 H/Y. Stat170 workshop on Hypothesis Testing, prepared by Nan Carter for Numeracy Centre, Macquarie University. Examples taken from various sources and adapted for Stat170. This document is made up from workshops that cover hypothesis testing about means and proportions: the hypotheses are about i. population mean, µ, using one random sample of observations, with value for σ ii. population proportion, π, using one random sample of observations iii. population mean, µ, using one random sample of observations, without value for σ iv. two population means and interested in the difference, µd = µ1- µ2, using one random sample of items and the two variables observed on each item v. two population means using two random samples of items, without values for σ1 , σ2 i. SAMPLING DISTRIBUTION of MEANS We start by working with distributions for which the standard deviation σ is known, AND we are interested in testing an hypothesis about the mean of the population using data. from a single random sample of items. NOTE 1. Recall that we need to keep in mind that we are working with two distributions: • the distribution of the individuals in the population from which we take a random sample; this distribution has mean μ and standard deviation σ. • the distribution of means of samples that are all the same size (n); this is the sampling distribution of means of samples of size n; this distribution has mean μ and standard deviation σ/√n (also referred to as the ‘standard error’). Notice that • The two distributions have the same mean μ • the means of samples are less variable than the individuals in the ‘parent’ population (the means have s.d. σ/√n where the individuals have s.d. σ) • if you take larger samples (i.e. n is larger) the means of these samples are even less variable (s.d. σ/√n gets smaller as n increases). {RECALL: the Central Limit Theorem, which proves that, if the parent distribution is NOT NORMAL (or NOT KNOWN) BUT the samples are large enough, then the sampling distribution of means is approximately normal. At Macquarie we can apply the CLT if n>25. THIS MEANS THAT, ONE WAY OR ANOTHER, WE CAN HAVE A SAMPLING DISTRIBUTION OF MEANS THAT IS NORMAL (APPROX at least).} NOTE 2. If we have a normal distribution with mean μ and standard deviation σ/√n, then 95% of the values in the distribution lie in the interval from (μ - 1.96 x σ/√n to μ + 1.96 x σ/√n). This means that, if we took lots and lots of samples and found the mean of each, we would expect that 95% of those means would lie in that interval. NOTE 3. IT ALSO MEANS THAT (think this through carefully): • if we do not know μ, but have a lot of estimates ybar from samples (all of the same size, n) and for each ybar we calculate the interval (ybar - 1.96 x σ/√n to ybar + 1.96 x σ/√n), then we expect that 95% of these intervals will include the true value of μ. IN OTHER WORDS: we can expect that the true value of μ will be included in 95% of the intervals. 15 Nan Carter: workshop notes prepared for Numeracy Centre Macquarie University.
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STAT170 Workshop Notes prepared by Nan Carter for Numeracy ...

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