5.4 Normal Approximation of the Binomial Distribution Lesson MDM4U Jensen

Transcription

1 5.4 Normal Approximation of the Binomial Distribution Lesson MDM4U Jensen Review From Yesterday Bernoulli Trials have 3 properties: Binomial Probability Distribution In a binomial experiment with n Bernoulli trials, each with a probability of success p, the probability of k successes in the n trials is given by: Where X is the discrete random variable corresponding to the number of successes. Warm- Up: a) What is the probability of getting exactly 30 heads if a coin is tossed 50 times? b) Determine the expected number of heads in 50 trials

2 c) Find the probability of tossing between 20 and 30 heads, inclusive, in 50 tosses of a coin. P(20 X 30) This is where the real problem occurs with more complex situations. You could calculate it with 11 such calculations: P(20 X 30) = P(X = 20) + P(X = 21) + P (X = 22) + P(X = 30) This calculation would be very time consuming! To help simplify the calculation, look at the graphical representation of the binomial distribution. A binomial distribution can be approximated by a normal distribution as long as the number of trials is large! - It is mound shaped and the normal curve is a close fit. - To use the normal curve, you must know values for the and because you must calculate the in order to translate the data from the normal curve to the standard normal curve. This will allow you to determine the percentage of data that has an equal or lower z- score. The percentage is equivalent to the area under the curve. What is the Standard Normal Curve? The standard normal distribution is a special normal distribution with a mean of and a standard deviation of.

3 Mean and Standard Deviation of the Normal Approximation to the Binomial Distribution Practice If you were to toss a coin 50 times, calculate. a) the mean ( x ) number of heads in 50 tosses b) the standard deviation ( σ ) of the number of heads in 50 tosses From Discrete to Continuous There is one major issue. - A binomial distribution represents a random variable - A normal distribution is - In order to use the normal distribution to approximate the binomial distribution, you must consider a range of values rather than specific discrete values. Example: To find a range of values to represent the discrete value of 5, you must include all numbers that round to 5. Therefore, the range of values between 4.5 and 5.5 can be used to represent the discrete value of 5.

4 Practice finding the range of continuous values for a discrete value. a) What range of continuous values can be used to represent P(X = 2)? b) What range of continuous values can be used to represent P(2 X 4)? c) What range of continuous values can be used to represent P(X > 3)? d) What range of continuous values can be used to represent P(X 3)?

5 e) What range of continuous values can be used to represent P(2 < X < 5)? f) What range of continuous values can be used to represent P(X < 11)? g) What range of continuous values can be used to represent P(X 7)? Example 1: Fran tosses a fair coin 50 times. Estimate the probability that she will get tails less than 20 times. This means you must find P(X < 20) where X is the number of tails in 50 tosses) Step 1: Find the mean and standard deviation

6 Step 2: Determine continuous values for P(X < 20) Step 3: Find the z- score Step 4: Use your z- score to determine the probability using the chart [Find P(z < )] Therefore there is a % chance that Fran will toss less than 20 tails in 50 attempts. Example 2: Calculate the probability that, in 100 rolls of a fair die, a 6 appears between 10 and 20 times, inclusive. [Find P(10 X 20)] Step 1: Find the mean and standard deviation Step 2: Find the continuous range for the discrete values of P(10 X 20)

7 Step 3: Find the z- scores Step 4: Use your z- scores to determine the probability using the chart Therefore the probability of rolling between 10 and 20 (inclusive) sixes on a fair die that is rolled 100 times is %. Example 2 Using Your Ti- 83 Calculator A TI- 83 calculator can be used to calculate the area under the normal distribution. Use the command: normalcdf(lower X value, upper X value, mean, stand. dev.) access the command by pressing: 2nd - - > distr - - > 2

8 Example 3: A hamburger patty producer claims that its burgers contain 400 grams of beef. It has been determined that 85% of burgers contain 400 grams or more. An inspector will only accept a shipment if at least 90% of a sample of 250 burgers contain more than 400 grams. What is the probability that a shipment is accepted? Step 1: Find the mean and standard deviation Step 2: Find the continuous range for the discrete values of P(X 225) Step 3: Find the z- score Step 4: Use your z- score to determine the probability using the chart There is a % chance that the shipment is accepted.

9 WARNING Not all binomial distributions can be approximated using normal distributions. Was a normal approximation of the binomial distribution appropriate for example 3? Since np > 5 and 1(n - p) > 5, the binomial distribution can be approximated by the normal curve.