Normal Curve approximations to Binomial Distributions

Normal Distribution Curve by flickr user AceTrace

Are the following distributions normal approximations of binomial distributions? How do you know?

(d) 80 trials where the probability of success on each trial is 0.99

(c) 600 trials where the probability of success on each trial is 0.05

(b) 60 trials where the probability of success on each trial is 0.20

(a) 60 trials where the probability of success on each trial is 0.05

Question #1HOMEWORK

We now want to use the normal approximation of a binomial distribution. The distribution will be approximately normal if:

Once we know that a binomial distribution can be approximated by a normal curve we can calculate the values of μ and σ like this:

np ≥ 5 and nq≥ 5

Normal Approximation to the Binomial Distribution

this is the LINK

between these two

types of distributions

Link by flickr user jontintinjordan

remember the LINK

Determine the mean and standard deviation for each binomial distribution. Assume that each distribution is a reasonable approximation to a normal distribution.

Question #2

(c) The probability of the Espro I engine failing in less than 50 000 km is 0.08. In 1998, 16 000 engines were produced. Find the mean and standard deviation for the engines that did not fail.

(b) 44 trials where the probability of failure for each trial is 0.28

(a) 50 trials where the probability of success for each trial is 0.35

A laboratory supply company breeds rats for lab testing. Assume that male and female rats are equally likely to be born.

(b) What is the probability that of 240 animals born, 110 or more will be female?

(a) What is the probability that of 240 animals born, exactly 110 will be female?

Question #3Solve the following problem using a binomial solution

A laboratory supply company breeds rats for lab testing. Assume that male and female rats are equally likely to be born.

(d) Is it correct to say that, in the above situation, P(x ≥ 120) = P(x > 119), or do we need to account for the values between 119 and 120?

(c) What is the probability that of 240 animals born, 120 or more will be female?

Question #3Solve the following problem using a binomial solution

Normal Approximation to the Binomial Distribution

We have seen that binomial distributions and their histograms are similar to normal distributions. In certain cases, a binomial distribution is a reasonable approximation of a normal distribution. How can we tell when this is true?

Confused by flickr user slava

Recall:

In a normal distribution, we used values for μ and σ to solve problems, where:

• μ = the population mean, and • σ = the standard deviation

In a binomial distribution, we used values for 'n' and 'p' to solve problems, where:

• n = number of trials, and • p = probability of success

Normal Approximation to the Binomial Distribution

We now want to use the normal approximation of a binomial distribution. The distribution will be approximately normal if:

Once we know that a binomial distribution can be approximated by a normal curve we can calculate the values of μ and σ like this:

np ≥ 5 and nq≥ 5

Normal Approximation to the Binomial Distribution

this is the LINK

between these two

types of distributions

Link by flickr user jontintinjordan

An Example

Border patrol officers estimate that 10 percent of the vehicles crossing the US - Canada border carry undeclared goods. One day the officers searched 350 randomly selected vehicles. What is the probability that 40 or more vehicles carried undeclared goods?

Normal Approximation to the Binomial Distribution

What is n?

What is p?

What is q?

What is μ?

What is σ?

Is this binomial distribution approximately normal?

Is np ≥ 5?

Is nq ≥ 5?

A laboratory supply company breeds rats for lab testing. Assume that male and female rats are equally likely to be born.

(c) Compare the above answers to #3b and #3c.

(b) What is the probability that of 240 animals born, 120 or more will be female?

(a) What is the probability that of 240 animals born, 110 or more will be female?

Question #4Solve the following binomial problem as normal distribution problem

The probability that a student owns a CD player is 3/5. If eight students are selected at random, what is the probability that:

(c) none of them own a CD player?

(b) all of them own a CD player?

(a) exactly four of them own a CD player?

HOMEWORK

The probability that a motorist will use a credit card for gas purchases at a large service station on the Trans Canada Highway is 7/8. If eight cars pull up to the gas pumps, what is the probability that:

(b) four of them will use a credit card?

(a) seven of them will use a credit card?

HOMEWORK