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A linear regression fits the lineor as Excel prefers to call itto the existing data set. It does so through a technique known as minimizing the sum of the squares of the error terms. To get the complete result of a regression analysis, select a range 5 rows by 2 columns and array-enter the LINEST function as shown in Figure 4.
The first row contains the 2 coefficients a1 and a0 respectively. The rest of the information is important in understanding how well the regression line fits the data, how significant the individual coefficients are, as well as the significance of the regression as a whole.
It also contains key elements needed to build confidence intervals for interpolated or extrapolated estimates, a subject covered in the section titled Confidence Intervals.
But, first, we start with some nomenclature.
The number of data points is given by n. The number of independent variables is given by k. If a constant is included in the regression, it increases k by 1.
Each of the recorded observations is denoted by the pair of values.
For eachthe value predicted by the regression is given by. The value that one gets from the regression is. For reasons that will soon be apparent, we start with the last row.
There are two values and. These are aggregate measures of something we have already looked at the level of an individual data point.
Recall that for each individual data point, the measure of how much the regression explains is and how much remains unexplained is. The first is the sum of the squared values of how well the regression fits the data or. The second is the sum of the squared values of how much remains unexplained or.
Row 4 contains two values: The degrees of freedom is given by the expression n-k, where n and k are explained earlier in this section. The F statistic, or the observed F-value, is a measure of the significance of the regression as a whole.
For the technically minded it tests the null hypothesis that all of the coefficients are insignificant against the alternative hypothesis that at least one of the coefficients is significant.
While Excel provides the value, it can also be computed as. This, the observed F-value, is then compared against a critical F-value, F a, v1, v2where a is 1 - the level of significance we are interested in, and v1 and v2 are as calculated below. If the observed F-value is greater than the critical F-value, it means the regression as a whole is significant.
Row 3 contains the two metrics, R2 and the SEreg. The R2 is measure of how well the regression fits the observed data.
It ranges from 0 to 1 and the closer to 1 the better the fit. Mathematically, it is calculated aswhere each term is explained above. Graphically, in terms of Figure 5, it is a measure of how close the regression line is to all of the observations. Suppose the regression line were to pass through every observation.
The second item in this row is the Standard Error of the regression, or SEreg. It can also be calculated from what we already know, i.
Keep in mind that n-k is also the df value in row 4 of the result. SSreg will play a role in calculating the confidence intervals later in this chapter.
Row 2 provides the standard error of each coefficient, or. The section Understanding the result addresses how these errors help determine if the coefficients are significant. Used as an array formula in a 5 rows by X columns range, LINEST returns not only the coefficients but also other statistical information about the results.Reversible reactions and dynamic equilibrium (AQA GCSE Chemistry paper 2, Topic 6 "The rate and extent of chemical change").
Topic 6 The rate and extent of chemical change QUIZ (AQA GCSE Chemistry). Reversible reactions. Know that in some chemical reactions, the products of the reaction can react to produce the original reactants.
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Directions: Write at least two linear equations so that the solution of the system of equations of that line and 4x + y = 8 is (3, -4) What does it mean to be a. Year 9 Term 3 Year 9 Term 2 Year 9 Term1 Summary Notes Wk No DfE Ref Resources a Four rules Use non-calculator methods to calculate the sum, difference, product and quotient of positive and negative whole numbers. Ask Math Questions you want answered Share your favorite Solution to a math problem Share a Story about your experiences with Math which could inspire or .
Directions: Write at least two linear equations so that the solution of the system of equations of that line and 4x + y = 8 is (3, -4) What does it mean to be a. Search the world's information, including webpages, images, videos and more.
Google has many special features to help you find exactly what you're looking for. Revision summaries for Paper 2 AQA GCSE Chemistry 2 (this page) (separate science AQA GCSE chemistry). What's assessed in this paper?
SUMMARY Topic 6 The rate and extent of chemical change (AQA GCSE Chemistry 2 paper 2). Topic Rate of reaction Topic Reversible reactions and dynamic equilibrium. SUMMARY Topic 7 . Choose at least one problem type below. graph a line (linear equation), given its equation in the form y = mx + b graph a line (linear equation), given its equation in the normal form (Ax + By + C = 0) graph a line (linear equation), given its slope and one point .