Teaching S1/S2 statistics using graphing technology

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2 Teaching S1/S2 statistics using graphing technology

3 CALCULATOR HINTS FOR S1 & 2 STATISTICS - STAT MENU (2) on Casio. It is advised that mean and standard deviation are obtained directly from a calculator. 1. Numerical measures Mean and standard deviation No frequencies List 1: input x F2 (CALC) 3

4 F6 SET 1 Var X List: List 1 F1 1 Var Freq : 1 EXIT F1 (1 VAR)

5 Frequencies List 1: input x List 2: input frequencies For grouped data, the mid values should be entered into List 1. This can be done by entering, for example, ( )/2 directly into List 1. F2 (CALC) F6 SET 1 Var X List : List 1 1 Var Freq : List 2 EXIT F1 (1 VAR) 5

6 1. Find the mode. 2. Find the mean. 3. Find the median.

7 It is advised that calculator is used to obtain probabilities. Binomial Distribution (a) Probabilities of type P(X = x): eg B( 15, 0.2) P( X = 3) F5 DIST F5 BINOMIAL F1 Bpd Data Variable x : 3 EXE Numtrial : 15 EXE p : 0.2 EXE Execute F1 (calc)

8 (b) Probabilities of type P(X x): eg B( 15, 0.2) P( X 3) F2 Bcd x : 3 EXE Numtrial : 15 EXE p : 0.2 EXE Execute F1 (calc)

9 AQA Jan 2007 Statistics 1B

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12 It is advised that students should be aware of the method involved as questions may involve being asked to show a result. Discrete Probability Distributions: Expectation and variance List 1: input x List 2: input probabilities CALC SET EXIT 1 VAR X x 2 x = E(X) = E(X) =E(X²) σ given Check n = 1 12

13 Example For the following probability distribution, x P (X = x) calculate (a) E(X) ; (b) E( X²) ; (c) the variance of X; (d) the standard deviation of X. 13

14 (a) E(X) =1.98 (b) E(X²) = 5.36 (c) Var = ² = 1.44 (d) sd =

15 It is advised that z values and methods are shown and the calculator is used for checking results. Normal Distribution: Calculation of probabilities (a) Probabilities of types P(X x), P(X < x), P(X > x) and P(X x) can be found directly eg X~ Normal mean 135 st deviation 15 P(X 127) or P(X < 127) F5 DIST F1 Norm F2 Ncd Select a suitable Lower value enter info

16 eg X~ Normal mean 135 st deviation 15 P(X 118) or P(X >118) Select a suitable Upper value - enter info eg X~ Normal mean 135 st deviation 15 P( 119 < X < 128)

17 (b) Problems involving inverse normal probabilities : F5 DIST F1 NORM F3 InvN eg X~ Normal mean 135 st deviation 15 Find value of x such that P(X< x) = 0.15 F3 Inv N Enter data Area Left z scores corresponding to area can also be obtained (as Inv Normal tables)

18 eg X~ Normal mean 135 st deviation 15 Find value of x such that P(X > x) = 0.30 Area right this time

19 AQA Jan 2011 Statistics 1B

20 (iii) 0 (i) (ii)

21 Correlation and regression It is advised that the calculator is used, if possible, to find the values of r, a and b. List 1: input x List 2: input y CALC F3 REG F1 X F2 a + bx This gives a,b,r (and r² ) for regression equation y = a + bx y = x

22 This can also be done from the GRAPH, Scatter Function F1 graph Calc X a +bx F1 graph 1 (default scatter) F1 calc will produce results Draw Residuals calculation can be set up Shift Setup Resid List

23 Example The following data were collected by a UK Gas supply company. It shows, for a house in London, the weekly gas consumption, y, in thousands of cubic feet, and the average outside temperature, x C. week x y (a) Calculate the regression line of gas consumption on average outside temperature. (b) Evaluate the residuals for the points where x = 3.0 and x = 7.8 (c) Find an estimate for the weekly gas consumption when the outside temperature is 5 C.

24 (a) Calculate the regression line of gas consumption on average outside temperature. (b) Evaluate the residuals for the points where x = 3.0 and x = C (c) Find an estimate for the weekly gas consumption when the outside temp is 5 C (a) equation is y = x (b) Residual x =3 Residual x =7.8 (c) Predict y when x = 5 y = x = 5.02

25 It is advised that calculator/tables are used. Discrete Probability Distributions: Poisson distribution All probabilities: MENU STAT Example Probabilities of type P(X = x): F5 (DIST) F6 F1 (POISN) F1 (Ppd) eg Poisson distribution with λ = 2.8 P( X = 4)

26 (b) Probabilities of type P(X x): F5 (DIST) F6 F1 (POISN) F2 (Pcd) Example Poisson distribution with λ = 2.8 P( X 3)

27 It is advised that the z values used and the method are shown, with the calculator used for checking. Confidence intervals for μ (known variance or large sample using z) eg A random sample of 12 packets of crisps with nominal weight 35g is obtained. The weights of such packets can be modelled by a normal distribution with standard deviation 3.5g. The weights of each of the packets in the sample are: Calculate a 95% confidence interval for the mean weight. F1 z F1 Data in list 1 95% ( 34.39, )

28 eg A random sample of 112 rods, that had mean 4.76 mm and standard deviation 1.46 mm, was obtained from a production process on a particular Friday. Calculate a 90% confidence interval for the mean length of rods produced on this Friday. Variable as mean and stdev given Interval ( 4.53, 4.99) 28

29 It is advised that the t values used and the method are shown, with the calculator used for checking. Confidence intervals for μ (random sample from a normal distribution with unknown standard deviation using t- distribution.) eg A company manufactures components for racing cars. A random sample of these components is taken from the production line during an afternoon shift. The lengths, in mm, of the components in this sample were: Assuming the lengths may be modelled by a normal distribution, calculate a 90% confidence interval for the mean length.

30 F2 t F1 1 sample Data in List 1 90% so the interval is (135.29, )

31 Summations given Standard deviation not known use t First find the mean and an estimate for the standard deviation 31

32 . It is advised that the method used for obtaining the test statistic is shown with the calculator used for checking. Hypothesis test on population mean based on sample from a normal distribution with known or unknown standard deviation using z or t- distribution Example List of data given z test H0 μ = 35 H1 μ > 35 5% level A random sample of 10 packets of crisps with nominal weight 35g is obtained. These weights can be modelled by a normal distribution with standard deviation known to be 1.5 g. The weights of each of the packets in the sample are : F3 test F1 z F1 I sample

33 H 1 is μ > 35 z = (test statistic) p = < 0.05 sig level X = sx not needed n = 10 Sig evidence to reject Ho and conclude that there is significant evidence to suggest that the mean weight of crisps is higher than 35g

34 Example List of data given t test H0 μ = 35 H1 μ 35 5% level A random sample of 10 packets of crisps with nominal weight 35g is obtained. These weights can be modelled by a normal distribution The weights of each of the packets in the sample are : F3 test F2 t F1 I sample

35 H 1 is 35 t = (test statistic) p = X < 0.05 sig level = Sig evidence to reject Ho and conclude S x = that the mean weight of crisps is not n = 10 35g ( appears to be higher ).

36 It is advised that the method for obtaining expected frequencies is shown and also the method used for obtaining the test statistic. Application of Contingency tables: use of Example A random sample of people was asked their opinion on a road scheme. The people lived in the road to be involved in the scheme, one road away from the scheme or 2 or more roads away from the scheme. The results are summarised in the following table. ( O i Ei E i 2 ) 2 Test, using the distribution and the 1% level of significance, whether opinion is independent of where a person lives.

37 F3 test F3 Chi F2 2 way F2 to insert table Set dimensions EXE Enter data and EXIT Exe test stat and p value

38 F6 Go to Mat B Mat B gives the expected values Ho No association H1 Association p value < 0.01 ( sig level) Reject Ho Significant evidence that opinion is associated with whether respondent lives in the road involved, one road away from the scheme or 2 or more roads away from the scheme

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