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Summary Statistics 2, MAT15403, wur
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A summary of the lecture slides and the book for statistics 2, a course given in the first and second years of the WUR bachelors.
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Summary: An Introduction to Statistical Methods and Data Analysis, ISBN: 9781305269477 Statistics 2 (MAT15403)
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Statistics 2 summaryPopulations are characterized by parameters, such as the mean, the median, the standard deviation
and the proportion. Methods for making inferences about parameters fall into two categories:
estimate the value of the parameter or test a hypothesis.
The point estimate of μ is sample mean y (unbiased estimator). For a large n, y will be normally
distributed with a mean μ and a standard error σ/√n (central limit theorem). An interval estimate of
μ is the probability of y falling in the interval with a level of confidence of 0.95 (for example). .95 is
also called the confidence coefficient.
y−μ 0
The student’s t-test can be used when σ is unknown. t= , with df= n-1, and as n increases, the
s/√ n
distribution of t approaches the distribution of z. The z-test is used for a known σ, with a df=inf. The
tα values can be found in table 2, when looking at the appropriate df. Confidence interval: y ± tα/2 *
s/√n.
A statistical test needs a null hypothesis and a research hypothesis. The test consists of 8 steps:
1. Null hypothesis (H0) and research hypothesis (Ha)
2. Test statistics T.S. (the decision to state whether or not the data support the research
hypothesis)
3. T-distribution under H0 (define df).
4. T-distribution under Ha
5. Type of population mean test
6. Outcome of test statistic t
7. Outcome of p-value
8. Conclusion in (non)statistical terms
A rejection region contains the values of y that support the research hypothesis and reject H0. Type 1
error: rejecting H0 when it is true (based on α). Type 2 error: accept H0 when it is false and Ha is true.
A one-tailed test has the rejection region located in only one tail of the y distribution (for example
when μ>520). For a two-tailed test, Ha: μ ≠ 520, and both tails of the rejection region are included.
Tests for a population mean can be right-tailed (Ha: μ>520), left-tailed (μ<520) or two-tailed (μ ≠
520).
The level of significance (p-value) is defined as the probability of obtaining a value of T.S. that is likely
to reject H0. If p<α, reject H0, and for p>α, fail to reject H0.
Two independent random variables can be normally distributed with μ= μ1-μ2 and σ 2 = σ12 + σ22. The
standard error is σ = √( σ12/n1 + σ12/n2).
For this two-independent t-test, equal variances can be assumed, giving the following:
Sp2 is the weighted average of the sample variances, to determine the unknown σ. For this, a
hypothesis test can also be conducted with a μ= μ1-μ2. The two samples need to be independent
and randomly selected. Dependence can also be the result of spatial correlation.
If there are no equal variances, another test can be performed.
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