MTH3003 PJJ SEM I 2015/2016

 ASSIGNMENT :25% Assignment 1 (10%) Assignment 2 (15%)  Mid exam :30% Part A (Objective) Part B (Subjective)  Final Exam: 40% Part A (Objective) Part B (Subjective - Short) Part C (Subjective – Long)

o Definition o Graphing

MEASURES OF CENTER - Arithmetic Mean or Average - Median - Mode Group and ungrouped data

Range Interquartile Range Variance Standard Deviation Group an ungrouped data

interpret Calculate  Q1, Q2 and Q3, IQR, Upper fence, lower fence, outlier

lower and upper quartiles (Q 1 and Q 3 ),The lower and upper quartiles (Q 1 and Q 3 ), can be calculated as follows: position of Q 1The position of Q 1 is 0.75(n + 1)0.25(n + 1) position of Q 3The position of Q 3 is once the measurements have been ordered. If the positions are not integers, find the quartiles by interpolation.

The prices ($) of 18 brands of walking shoes: Position of Q 1 = 0.25(18 + 1) = 4.75 Position of Q 3 = 0.75(18 + 1) = Example

Basic concept The probability of an event - how to find prob Counting rules Calculate probabilities

 Event Relations: Union, Intersection, Complement  Calculating Probabilities for Unions The Additive Rule for Unions A Special Case – Mutually Exclusive Complements Intersections Independent and Dependent Events Conditional Probabilities The Multiplicative Rule for Intersections

Probability Distributions for Discrete Random Variables  Properties for Discrete Random Variables  Expected Value and Variance

 The properties for a discrete probability function (PMF) are:  Cumulative Distribution Function (CDF)

 Toss a fair coin three times and define X = number of heads. 1/8 P(X = 0) = 1/8 P(X = 1) = 3/8 P(X = 2) = 3/8 P(X = 3) = 1/8 P(X = 0) = 1/8 P(X = 1) = 3/8 P(X = 2) = 3/8 P(X = 3) = 1/8 HHH HHT HTH THH HTT THT TTH TTT x x Xp( x ) 01/8 13/8 2 31/8

Discrete distributions: binomial The binomial distribution Poisson The Poisson distribution hypergeometric The hypergeometric distribution  To find probabilities formula cumulative table

I. The Binomial Random Variable 1. Five characteristics: n identical independent trials, each resulting in either success S or failure F; probability of success is p and remains constant from trial to trial; and x is the number of successes in n trials. 2. Calculating binomial probabilities a. Formula: b. Cumulative binomial tables 3. Mean of the binomial random variable:   np 4. Variance and standard deviation:  2  npq and

A marksman hits a target 80% of the time. He fires five shots at the target. What is the probability that exactly 3 shots hit the target? P(x = 3) P(x = 3) = P(x  3) – P(x  2) = =.205 P(x = 3) P(x = 3) = P(x  3) – P(x  2) = =.205 Check from formula: P(x = 3) =.205

II. The Poisson Random Variable Examples: 1. The number of events that occur in a period of time or space, during which an average of  such events are expected to occur. Examples: The number of calls received by a switchboard during a given period of time. The number of machine breakdowns in a day 2. Calculating Poisson probabilities a. Formula: b. Cumulative Poisson tables 3. Mean of the Poisson random variable: E(x)  4. Variance and standard deviation:  2   and

III. The Hypergeometric Random Variable 1. The number of successes in a sample of size n from a finite population containing M successes and N  M failures 2. Formula for the probability of k successes in n trials: 3. Mean of the hypergeometric random variable: 4. Variance and standard deviation:

A package of 8 AA batteries contains 2 batteries that are defective. A student randomly selects four batteries and replaces the batteries in his calculator. What is the probability that all four batteries work? Success = working battery N = 8 M = 6 n = 4

The Standard Normal Distribution 1. The normal random variable z has mean 0 and standard deviation Any normal random variable x can be transformed to a standard normal random variable using 3. Convert necessary values of x to z. 4. Use Normal Table to compute standard normal probabilities.

The weights of packages of ground beef are normally distributed with mean 1 pound and standard deviation 0.1. What is the probability that a randomly selected package weighs between 0.80 and 0.85 pounds?

 We can calculate binomial probabilities using  The binomial formula  The cumulative binomial tables  When n is large, and p is not too close to zero or one, areas under the normal curve with mean np and variance npq can be used to approximate binomial probabilities.

continuity correction. Make sure to include the entire rectangle for the values of x in the interval of interest. That is, correct the value of x by This is called the continuity correction. Standardize the values of x using Make sure that np and nq are both greater than 5 to avoid inaccurate approximations!

Suppose x is a binomial random variable with n = 30 and p =.4. Using the normal approximation to find P(x  10). n = 30 p =.4 q =.6 np = 12nq = 18 The normal approximation is ok!

 Sampling Distributions  Sampling distribution of the sample mean  Sampling distribution of a sample proportion  Finding Probabilities for the  Sample Mean  Sample Proportion

A random sample of size n is selected from a population with mean  and standard deviation   he sampling distribution of the sample mean will have mean  and standard deviation. normal, If the original population is normal, the sampling distribution will be normal for any sample size. non normal, If the original population is non normal, the sampling distribution will be normal when n is large. The standard deviation of x-bar is sometimes called the STANDARD ERROR (SE).

If the sampling distribution of is normal or approximately normal  standardize or rescale the interval of interest in terms of Find the appropriate area using Z Table. If the sampling distribution of is normal or approximately normal  standardize or rescale the interval of interest in terms of Find the appropriate area using Z Table. Example: Example: A random sample of size n = 16 from a normal distribution with  = 10 and  = 8.

The standard deviation of p-hat is sometimes called the STANDARD ERROR (SE) of p-hat. A random sample of size n is selected from a binomial population with parameter p. The sampling distribution of the sample proportion, will have mean p and standard deviation approximately normal. If n is large, and p is not too close to zero or one, the sampling distribution of will be approximately normal.

Example: Example: A random sample of size n = 100 from a binomial population with p = 0.4. If the sampling distribution of is normal or approximately normal, standardize or rescale the interval of interest in terms of Find the appropriate area using Z Table. If the sampling distribution of is normal or approximately normal, standardize or rescale the interval of interest in terms of Find the appropriate area using Z Table. If both np > 5 and np(1-p) > 5