#MEIConf2018. Before the age of the Calculator

Transcription

1 @MEIConference

2 Before the age of the Calculator

3 Since the age of the Calculator

4 New A Level Specifications To use technology such as calculators and computers effectively

5 Session Aims: To use different menus to investigate Normal distribution properties. To generate tables using SEQUENCES To create Binomial graphs to investigate the effects of the parameter p

6 Using the Run and Stats Menus Considering the Standard Normal Z~N(0,1 2 )

7 What z value gives p = 0.2? Show this on a graph.

8 Using Sequences in Tables

9 Creating Tables and Graphs for the Binomial Distribution:

10 About MEI Registered charity committed to improving mathematics education Independent UK curriculum development body We offer continuing professional development courses, provide specialist tuition for students and work with employers to enhance mathematical skills in the workplace We also pioneer the development of innovative teaching and learning resources

11 Instruction sheet 1: Exploring the area properties of the Standard Normal Distribution. The Standard Normal distribution is a set of data where the mean is 0 and the standard deviation is 1. The notation for a Standard Normal Distribution is Z~N(0,1 2 ). 1. Go into the Run/Matrix mode. Press 1 on the menu screen. 2. Press iy to enter the Statistics options and then e to select Distributions. 3. We want the Inverse Normal function so press q for Normal and e for the Inverse function. 4. Enter the values 0.2, 1, 0 to find the inverse value and store this value in A 0.2,1,0klbafl 5. Go into the Statistics mode. Press p2. Enter the information we have just found into the Normal distribution calculator and show the graph of our findings: 6. Press y to select Distributions, q for Normal and w for Cumulative Normal 7. Select w to enter the data. 8. Enter the lower limit as -9999, upper limit as the value found in 4. and then draw the graph (this will pop up as an option in the bottom right corner when you go to the Execute row). Nn9999laflNNNNu

12 Further Investigation Questions: 1. Find a such that P(Z < a) = 0.3, show this in a graph. 2. Find 3 combinations of b and c such that P(b < Z < c) = 0.3. Sketch them. 3. If event A is P(Z < d) and event B is P(Z > e) find 3 combinations of d and e such that P(A B) = 0.3. Sketch them. Problem: A random variable X is distributed Normally with mean of 20 and variance 9. a) Find ( ) b) Let ( ) i. Find the value of k. ii. Represent this information on a copy of the diagram below. Further Tasks: Consider the Normally distributed variable X with mean 20 and variance 2. If you know that ( ), how could you calculate? What information would you need if you didn t know either or?

13 Instruction sheet 2: Exploring the shape of Binomial pdf and cdf curves. What are the conditions for a variable X to be Binomially distributed? The Binomial distribution is a set of data, X, where X represents the number of successes in a fixed number of trials, n. Each event is independent, there are only 2 possible outcomes and the probability of a success is p. The notation for a Binomial Distribution is X~B(n,p). Setting up the probability distribution X~B(30,0.5): 1. Go into the Statistics mode. Press 2 on the menu screen. 2. To auto fill the X values press BB and move the highlighted cell to the words List 1. From the options menu, select List : iq and enter the formula: Seq(x,x,0,30,1) (This will enter the numbers from 0 to 30 in steps of 1) yf,f,0,30,1kl 3. Next, return to the original menu dd and create the P(X=x) in the next column. Go to Distributions y, Binomial y and pdf q. Select List in order to create a table of results q 4. Move down to fill in the number of trials and probability value and store the outcomes in List 2 NN30l0.5lw2ll 5. Now return to the original table dd and draw the graph of the results. Press Graph q and select Graph 1 q. Note: you can alter the graphing options using SET on Graph options u

14 Further Investigation Questions: 1. Create a distribution X~B(30,p) that has a positive skew. 2. Create a distribution X~B(30,p) that has a negative skew. 3. Create a cdf, P(X x) with X~B(30,p) such that P(X 6) 0.5, giving p to 2 dp. Note: You will need to create a column of 0s for the lower limit of the cumulative function. You can trace along the values in the graph by pressing Lq Problem: In a sack containing a large number of beads are coloured gold and the remainder are of different colours. A group of children use some of the beads in a craft lesson and do not replace them. Afterwards the teacher wishes to know whether or not the proportion of gold beads left in the sack has changed. He selects a random sample of 20 beads and finds that 2 of them are coloured gold. Test, at the 10% level of significance, whether or not there is evidence that the proportion of gold beads has reduced. Justify your conclusion. Further Tasks: Investigate the conditions for when the Binomially Distributed X, X~B(n,p), can be approximated with the Normal Distribution.