Transcript Section 6-6

Section 6-6
Normal as Approximation to Binomial
Remember
Binomial Probability Distribution
1. There must be a fixed number of trials.
2. Each trial must be independent.
3. Every trial has exactly two possible
outcomes.
4. The probability of each outcome remains
consistent throughout every trial.
New Notes for Binomials
When both 𝑛𝑝 ≥ 5 and 𝑛𝑞 ≥ 5, then a binomial
can be treated as a normal distribution with the
mean 𝜇 = 𝑛𝑝 and 𝜎 = 𝑛𝑝𝑞.
Remember:
• p is probability of success
• q is probability of failure (1-p)
• n is number of trials
Continuity Correction
• X represents the number of successes which
is discrete, however normal distributions
are for continuous random variables.
• As a result, we must do a “continuity
correction”, in which we create a 1 unit
interval around x, by considering x - 0.5 and
x + 0.5
The How To
For x < 120: Calculate the area to the left of 119.5
For x < 120: Calculate the area to the left of 120.5
The How To
For x > 120: Calculate the area to the right of 120.5
For x > 120: Calculate the area to the right of 119.5
The How To
For x = 120: Calculate the area between
119.5 and 120.5
Overall Process
• Given Binomial:
1. Determine if it can be approximated as a normal
by checking if np and nq are > 5.
2. Find 𝜇 = 𝑛𝑝 and 𝜎 = 𝑛𝑝𝑞
3. Identify the value of X and determine the
appropriate continuity correction
4. Calculate the z Score for the value(s) found in
step 3 (using 𝜇 and 𝜎 from step 2).
5. Use table to find probability
Example 1
• If a gender-selection technique is tested
with 500 couples that have 1 baby, find
the probability of getting at least 275 girls.
Solution
1. This can be approximated using normal
because 𝑛𝑝 = 𝑛𝑞 = 500(.5) = 250 ≥ 5.
2. 𝜇 = 𝑛𝑝 = 500 .5 = 250,
𝜎 = 𝑛𝑝𝑞 = 500 .5 (.5) = 125 = 11.18
3. Since “at least” means ≥ we want to find the
area to the right of 275 – 0.5 = 274.5
4. 𝑧 =
𝑥−𝜇
𝜎
=
274.5−250
11.18
= 2.19
Solution (Continued)
5. The area to the left of 𝑧 = 2.19 is .9857, so the
area to the right is 𝐴 = 1 − .9857 = .0143
6. The conclusion is that the probability of 273
female births out of 500 is .0143.
Example 2
• Suppose that a sample of 20 tires of the
same type are obtained at random. It is
understood that 8% of all the tires are
defective. What is the probability that 15
or fewer of the collected tires are
defective?
Example 3
• If 10% of men are bald, what is the
probability of selecting less than 100 bald
men in a sample of 818 men.
Example 4
• 62% of 12th graders living in Oswego
attend OHS. If a sample of 500 12th
graders from the city are selected, find the
probability that more than 290 of them go
to OHS.
Example 5
• If you flip a coin 20 times, what is the
probability of it landing on heads exactly
10 times?
Homework
Pg. 306-307
#5-9,14-16, 21(a, b), 27
Homework Answers
Pg. 306-307 #5-9,14-16, 21(a, b), 27
5. The area to the right of 8.5.
6. The area to the right of 1.5.
7. The area to the left of 4.5.
8. The area between 3.5 and 4.5.
9. The area to the left of 15.5.
14. normal approx. is not suitable
15. normal approx. is not suitable
16. Normal approx: 0.0329
21. a.) 0.0318 b.) 0.2676
27. 0.2776. No, 27 blue M&Ms is not unusually
high because 0.2776 is not small.
HOMEWORK QUIZ:
• 62% of 12th graders living in Oswego
attend OHS. If a sample of 500 12th
graders from the city are selected, find the
probability that more than 290 of them go
to OHS.