PED lecture notes

# 2 statistical hypothesis reflecting the statistics

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2. Statistical hypothesis reflecting the statistics used to summarize your observations. e.g., ρ > 0: There is a positive correlation between motivation to learn and math achievement. The statistic of correlation is used to summarize data. μ g > μ b : Mean math achievement of girls is higher than the mean of boys. Mean is used to summarize data. 3. Null hypothesis representing a way to test the statistical hypothesis. μ g = μ b . The mean math achievement of girls is the same as the mean of boys. ρ = 0. There is no correlation between motivation to learn and math achievement. 4. Statistical tests are conducted with the assumption that the null hypothesis is true. What is the probability of finding a positive correlation when the truth is there is no correlation? What is the probability of finding a difference between the two means when there is

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no difference? Statistical Significance The probability level at which you will reject the null hypothesis, or, at which you will allow yourself the risk of wrongly rejecting the null hypothesis. Type I Error Significance level is also Type I error rate. It is the probability of rejecting the null hypothesis when the null hypothesis is true. You make such an error only when the null is rejected. Type II Error It is the probability of not rejecting the null hypothesis when the null hypothesis is false. You make such an error only when you fail to reject the null hypothesis. Sampling distribution of means (or any statistic) Is an imagined or theoretical distribution of an infinite number of means computed from random samples of the same size. Because of the central limit theorem, this distribution is used as a probability distribution to determine the probability of obtaining a mean larger than or as large as (in absolute value) the one computed from your sample. Central limit theorem 1. If repeated random samples of size n are drawn from a normally distributed population, the distribution of the sample means is normal. 2. As the sample size increases, disregarding the shape of the population distribution, the sampling distribution of means approximates normality. 3. The mean of the sampling distribution of means equals the population mean. 4. The standard deviation of the sampling distribution of means equals the population standard deviation divided by the square root of sample size. This is called standard error of means. If population variance is not known, sample variance can be used as an estimate of population variance in computing the standard error. Four steps in hypothesis testing: Example 1: State the null and H 0 : μ 1 2 = 0 alternative hypotheses. H 1 : μ 1 2 > 0 2: Set the level of statistical significance α .05 which is the probability at which you'll t (.05, 28)=1.7 reject the null or at which you'll allow yourself to make the type I error.
• Fall '11
• John Smith
• external validity

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