Presentation on theme: "Chapter 8 The shape of data: probability distributions"— Presentation transcript:
1Chapter 8 The shape of data: probability distributions
2The Binomial Distribution Is a discrete probability distribution and is appropriate when:A variable can only take on one of two valuesThe probability of the two outcomes are constant from trial to trailSuccessive events are independent
3Binomial formula The formula for the binomial distribution is Where nCr =n is the number of trials and r is the number of successes
4ExampleThe probability that an invoice will be returned because of an error is 0.1. If there are 20 invoices what is the probability that(a) exactly 2 invoices will be returned(b) at least 2 invoices will be returned
5(a) Probability of exactly two invoices in error = 190P(r = 2) = .12.918= 0.285
6(b) Probability of at least two invoices will be returned P(r > 2) = 1 – [P(r = 0) + P(r = 1) + P(r = 2)]P(r = 0) = 1 .10 .920=P(r = 1) = 20 .11 .919=P(r > 2) = 1 – ( )=
7Mean and standard deviation of the binomial distribution The mean of a binomial distribution is np.The standard deviation is given by the formula:
8The Poisson Distribution Another discrete probability distributionIt is good at modelling events that occur at random (e.g. arrivals at a supermarket checkout).The formula is:Where r is the number of events occurring in a given unit (of time or length etc.) and m is the mean number of events in the same unit and e is the constant …
9ExampleVisitors to a museum arrive at random with a mean of 2.5 per minute. What is the probability that there will beNo visitors in a one minute interval?Less than 2 visitors in a 2 minute interval?
10No visitors in 1 minuteP(0) ==(b) Less than 2 visitors in 2 minutesm = 2 2.5 = 5.0P(r<2) = P(0) + P(1)P(0) = e-5 =P(1) = 5=
11Mean and standard deviation of a Poisson distribution The mean is m and the variance is equal to the mean. So the standard deviation, which is the square root of the variance is equal to the square root of the mean. In symbols this becomes:
12Using the Poisson distribution as an approximation to the binomial distribution The number of trials, n, is large (greater than 30).The probability of a success, p, is small (less than 0.1).The mean number of successes, n p, is less than 5.
17The standard normal distribution Has a mean of 0 and a standard deviation of 1
18The normal tablesTables are used to solve normal distribution problems
19Area between Z of 1 and -1 P(Z>1) = 0.1587 P(Z<-1) = 0.1587 P(-1<Z<1) = 1 – 2 x=Or about 68%
20Z value if upper tail is 5% 5% represents a probability of 0.05Using tables in reverse we find that a Z value of 1.64 gives a probability of and a Z value of 1.65 gives a probability ofTaking an average gives us 1.645
21ExampleThe weight of a standard loaf is normally distributed with a mean of 800g and a standard deviation of 10g.Find the proportion of loaves that weigh more than 815 gThe baker wishes to ensure that no more than 5% of loaves are less than a certain weight. What is this weight?