Deskription. Exempel 1. Exempel 1 (lösning) Normalfördelningsmodellen (forts.)

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1 Deskription Normalfördelningsmodellen (forts.) 1 Exempel 1 En datorleverantör har en stödfunktion dit kunder med krånglande datorer kan ringa. Tiden det tar att svara på inkommande samtal varierar, och efter en längre tids mätningar har det visat sig rimlig att beskriva väntetiden som normalfördelad med genomsnittet 8 minuter och standardavvikelse 1.5 minuter. Hur stor andel av samtalen kommer under dessa förutsättningar att ta längre än 11 minuter? 2 Exempel 1 (lösning) 3 1

2 Exempel 2 Den ansvariga för faktureringen har tittat på resultaten från de senaste åren. Hon tycker sig se att försäljningen varje månad av en av företagets produkter kan beskrivas med en normalfördelning med genomsnittet kr och standardavvikelsen kr. Under hur stor andel av månaderna kommer försäljningen att understiga kr? 4 Exempel 2 (lösning) 5 Exempel 3 Det har visat sig någorlunda rimligt att beskriva den mängd socker som fylls i sockerpaketen med en normalfördelning där genomsnittsmängden är 2005 gram och standardavvikelsen är 5 gram. Vad blir högsta vikt för de 5% lättaste sockerpaketen? 6 2

3 Exempel 3 (lösning) 7 Scatterplots & Correlation 8 Examining relationship between two quantitative variables Explanatory and response variables Scatterplots and interpreting, outliers Categorical variables in scatterplots Quantifying linear relationships with correlation coefficient r Properties of correlation coefficient 3

4 Examining Relationships Most statistical studies involve more than one variable. Questions: What type of individuals do the data describe? What variables are present and how are they measured? Are all of the variables quantitative? Do some of the variables explain/cause changes in other variables? Relationships between two variables Most models are linear 1. Probabilistic models: Eg. Real estate prices in Luleå may be related to population per km 2 in the local area plus some random variation 2. Deterministic models: Eg: in electric current theory V=IR (unless for measurement errors, valtage of a given wire is proportaional to current flow) Most models in social sciences, economics, etc. may be probabilistic and often linear too! Start with a graph Looking at relationships Look for an overall pattern and deviations from the pattern Use numerical descriptions of the data and overall pattern (if appropriate) 4

5 Explanatory and response variables A response variable measures or records an outcome of a study. Also called dependent variable. An explanatory variable explains changes in the response variable (also called independent variable). response variable: real estate price explanatory variable: population per Km 2 Scatterplot A scatterplot shows the relationship between two quantitative variables measured on the same individuals. Typically, the explanatory or independent variable is plotted on the x axis, and the response or dependent variable is plotted on the y axis. Each individual in the data appears as a point in the plot. Scatterplot example Student Beers BAC

6 Interpreting scatterplots After plotting two variables on a scatterplot, we describe the relationship by examining the form, direction, and strength of the association. We look for an overall pattern Form: linear, curved, clusters, no pattern Direction: positive, negative, no direction Strength: how closely the points fit the form and deviations from that pattern. Outliers Form and direction of an association Linear relationships No relationship Some nonlinear relationships Positive association: High values of one variable tend to occur together with high values of the other variable. Negative association: High values of one variable tend to occur together with low values of the other variable. Following are very exact (deterministic) relationships 6

7 No relationship: X and Y vary independently. Knowing X tells you nothing about Y. Strength of the association The strength of the relationship between the two variables can be seen by how much variation, or scatter, there is around the main form. With a strong linear relationship, you can get a pretty good estimate of y if you know x. With a weak linear relationship, for any x you might get a wide range of y values. This is a weak relationship. For a particular state median household income, you can t predict the state per capita income very well. This is a very strong relationship. The daily amount of gas consumed can be predicted quite accurately for a given temperature value. 7

8 Outliers An outlier is a data value that has a low probability of occurrence (i.e., it is unusual or unexpected). In a scatterplot, outliers are points that fall (far) outside of the overall pattern of the relationship. Categorical variables in scatterplots Often, things are not simple and one dimensional. We need to group the data into categories to reveal trends. What may look like a positive linear relationship is in fact a series of negative linear associations. Plotting different habitats in different colors allows us to make that important distinction. Categorical explanatory variables When the explanatory variable is categorical, you cannot make a scatterplot, but you can compare the different categories side byside on the same graph (boxplots, or mean +/ standard deviation). Comparison of income (quantitative response variable) for different education levels (five categories). But be careful in your interpretation: This is NOT a positive association, because education is not quantitative. 8

9 Stronger association? Two scatterplots of the same data. The straightline pattern in the lower plot appears stronger because of the surrounding open space. The correlation coefficient r The correlation coefficient is a measure of the direction and strength of a linear relationship. It is calculated using the mean and the standard deviation of both the x and y variables. Correlation can only be used to describe quantitative variables. Categorical variables don t have means and standard deviations. How to calculate r? A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables. Linear relations are important because a straight line is a simple pattern that is quite common. Our eyes are not good judges of how strong a relationship is. Therefore, we use a numerical measure to supplement our scatterplot and help us interpret the strength of the linear relationship. The correlation r between x and y measures the strength of the linear relationship between two quantitative variables x and : r 1 x i x y i y n 1 s x s y 27 9

10 The correlation r between x and y measures the strength of the linear relationship between two quantitative variables x and : r 1 n 1 x i x s x y i y s y Ex. Ex. (forts) s y b1 r s x y b b x b y b x Facts about correlation r ignores the distinction between response and explanatory variables r measures the strength and direction of a linear relationship between two quantitative variables r is not affected by changes in the unit of measurement Positive value of r means association between the two variables is positive Negative value of r means association between the variables is negative r is always between 1 and +1 r is strongly affected by outliers 10

11 r ranges from 1 to +1 Strength: how closely the points follow a straight line. Direction: is positive when individuals with higher X values tend to have higher values of Y. When variability in one or both variables decreases, the correlation coefficient gets stronger ( closer to +1 or -1). Correlation only describes linear relationships No matter how strong the association, r does not describe curved relationships. 11

12 Influential points Correlations are calculated using means and standard deviations, and thus are NOT resistant to outliers. Just moving one point away from the general trend here decreases the correlation from to Correlation tells us about strength (scatter) and direction of the linear relationship between two quantitative variables. In addition, we would like to have a numerical description of how both variables vary together. And we would like to make predictions based on that numerical description (numerical model or equation). But which line best describes our data? Here we will ask Minitab to decide 12