Review of Central Tendency & Dispersion

Central Tendency & dispersion Mean 𝑥 = 1 𝑛 × 𝑖=1 𝑛 𝑥 𝑖 Median – Mid value after ranking Range Quartiles Range between Q1-Q3 capture what % of data? Calculated around the Median or Mean?

Dispersion – Cont… Variance – 𝑠 2 = 𝑖=1 𝑛 ( 𝑥 𝑖 − 𝑥 ) 2 𝑛−1 What is a drawback of using variance to describe dispersion? Standard Deviation - 𝑠= 𝑖=1 𝑛 ( 𝑥 𝑖 − 𝑥 ) 2 𝑛−1 What % of a sample is described by 2s? Which sample is more Variable?

Population statistics Parameter Mean 𝑥 µ Variance 𝑠 2 𝜎 2 Standard deviation 𝑠 𝜎

Exercise Provide the range, and 1st and 3rd quartiles for your dataset Manually calculate in excel the standard deviation for your sample of soil values. Using the stdev formula in excel calculate the standard deviations for all the sample sets How variable are the standard deviations? Which of the datasets comes closest to the population variance? (70.24 g.kg-1)

Hypotheses

Hypothesis A statistical hypothesis is a hypothesis that is testable on the basis of observing a process that is modelled via a set of random variables. Commonly two statistical datasets are compared, or a dataset obtained by sampling is compared against a synthetic dataset from an idealized model.

Hypothesis “a proposed explanation for a phenomenon.” (Wikipedia) Specifically a scientific hypothesis requires that the scientific method can be applied to test it.

Hypothesis Variety of test are available: Selection of the test will be based upon: The question / hypothesis to be tested. The data available to perform the tests. The assumptions of the tests and whether the data available meets the requirements.

Null & Alternative Hypothesis A hypothesis is proposed for the statistical relationship between two datasets , and this is compared as an alternative to an idealised null hypothesis that proposes no relationship between the two datasets. The Null hypothesis (Ho) states no difference The Alternate hypothesis (Ha) states the expected difference

Hypothesis example 1 “Because of high reflectance in the NIR region of the spectrum when looking at green vegetation I expect the mean reflectance value of healthy vegetation to be higher than urban structures.” Ha: There is no difference in the mean reflectance of vegetation and urban settlements in the NIR region of the spectrum Ho: Vegetation spectra have a higher reflectance in the NIR region c.f. Urban spectra

Is normal body temperature really 98.6 degrees F? Or it is lower? Example 2 Is normal body temperature really 98.6 degrees F? Or it is lower? The researcher starts assuming that the average adult body temp was 98.6 F degrees. The researcher selects a random sample of 130 adults. The average body temp. of the 130 sampled adults is 98.25. Make a decision about this assumption: likely or unlikely. Likely  the researcher does not reject this initial assumption (there is not enough evidence to do otherwise) Unlikely  either the researcher’s initial assumption is correct and he experienced a very unusual event or the researcher initial assumption is incorrect.

Example 3 Our criminal justice system assumes “the defendant is innocent until proven guilty” In the practice of statistics, we make our initial assumption when we state our two competing hypothesis (the null hypothesis (Ho) and the alternative hypothesis (Ha). The defendant is not guilty (Ho). The defendant is guilty (Ha) In statistics we always assume that the null hypothesis is true. If the jury finds sufficient evidence to make the assumption of innocence refutable. The jury rejects the null hypothesis and deems the defendant guilty. If there is insufficient evidence, then the jury does not reject the null hypothesis.

Errors in Hypothesis testing We make our decision based on the evidence not on 100% guaranteed proof. If we reject the null hypothesis, we do not prove that the alternative hypothesis is true. If we do not reject the null hypothesis, we do not prove that the null hypothesis is true.

Errors in Hypothesis testing We merely state that there is enough evidence to behave one way or the other. In statistics there is always a chance that we make an error. Two types of errors: Type I error: The null hypothesis is rejected when it is true. Type II error: The null hypothesis is not rejected when it is false.

Errors in Hypothesis testing Lets review the two types of errors that can be made in criminal trials: Truth Jury Decision Not guilty Guilty OK Error

Errors in Hypothesis testing Correspondence to the two types of errors in the hypothesis testing. Truth Decision Null Hypothesis Alternative Hypothesis Do not reject null OK Error II Reject null Error I Two types of errors: Type I error: The null hypothesis is rejected when it is true. Type II error: The null hypothesis is not rejected when it is false.

Making a decision It is likely or unlikely that we would observe the evidence we did given to the initial assumption. If it is likely, we don’t reject the null hypothesis. If it is unlikely, we reject the null hypothesis in favor of an alternative hypothesis.

Making a decision Look at each of the following hypothesis tests about the population mean. Type Null Alternative Right - tailed H0 : μ = 3 HA : μ > 3 Left- tailed HA : μ < 3 Two- tailed HA : μ ≠ 3 In practice: We would want to conduct the first hypothesis test if we were interested in conduct that the average grade point of the group is more than 3. We would want to conduct the second hypothesis if we were interested in concluding that the average grade point of the group is less than 3. We would want to conduct the third hypothesis test if we were only interested in concluding that the average grade point of the group differs from 3 (without caring whether it is more or less than 3)

Exercise Generate a couple of sensible hypotheses based on the population of Windhoek (either Whk on its own or comparison to the other Namibian cities) E.g. related to employment, or mode of transport, or education Generate a coupe of sensible hypotheses based on NUST student population E.g. related to evening and daytime students, or graduate and post-graduate, and current employment status

References References: http://www.r-tutor.com http://www.cyclismo.org/tutorial/R/types.html https://www.ijsr.net/archive/v3i8/MDIwMTU0ODU=.pdf Chun and Griffith (2013). Spatial Statistics and Geostatistics (Book)