Download Statistics Notes: 6.2 Binomal Probability Distributions

Statistics Notes: 6.2 Binomal Probability Distributions
6.2 The Binomial Probability Distribution
Binomial Probability Distribution: A probability distribution in which there are exactly two mutually exclusive outcomes. Criteria for a binomial probability experiment:
1. The experiment is performed a fixed number (n) of times (trials).
2. The trials are independent.
3. For each trial, there are 2 mutually exclusive outcomes:
• Success P(success) = p
P(success) = 0.7
• Failure P(failure) = 1 ­ p P(failure) = 0.3
4. The P(success) is fixed for each trial of the experiment.
Which of the following are binomial experiments? Check all 4 parts of the criteria.
A. A basketball player, who historically makes 80% of her free throws, is asked to shoot 3 free throws and the number of made free throws is recorded.
B. The number of people with blood type O­negative, based on a simple random sample of size 10 is recorded. According to the Information Please Almanac, 6% of the human population is O­negative.
C. 3 cards are drawn from a standard deck of playing cards. The number of aces drawn is recorded.
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Statistics Notes: 6.2 Binomal Probability Distributions
Binomial Probability Distribution Function: The probability of obtaining x successes in n independent trials of a binomial experiment is: According to Nielson, 75% of all US households have cable TV. In a random sample of 15 households, find:
A. the probability that exactly 10 have cable TV.
B. the probability that at least 13 have cable TV.
C. the probability that fewer than 13 have cable TV.
Mean and Standard Deviation of a Binomial Random Variable: According to Nielson, 75% of all US households have cable TV. Find the mean and standard deviation for a random sample of 300 households.
As the number of trials of a binomial experiment increases, the probability distribution of the random variable becomes normal. In general, if the variance is greater than 10, then the distribution will be bell shaped. x
68%
95%
99.7%
According to Nielson, 75% of all US households have cable TV. In a random sample of 300 households, 244 had cable. Is this unusual?
95% of the data is between 210 & 240
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Statistics Notes: 6.2 Binomal Probability Distributions
In 1998, 75% of all US households had cable TV. Yolanda and Lorrie think this percentage may have increased since 1998.
Yolanda conducts a simple random sample of 40 households and finds that 33 of them have cable. Assuming that 75% of the population had cable TV, compute the probability that 33 or more have cable TV. Does Yolanda's sample suggest that 75% is too low?
18.2% is not ≤ 5%, so this is not an unusual sample.
Yolanda has no evidence to suggest that 75% is too low. Lorrie conducts her own simple random sample of 40 households and finds that 38 have cable. Assuming 75% have cable, what is the probability that at least 38 have cable? Does Lorrie's sample support her belief that 75% is too low?
0.1% ≤ 5%, so this is an unusual sample.
Lorrie's sample suggests that 75% is too low. What does 0.0010 mean? or 1 random sample in 1000 will have either 38 or 39 or 40 households with cable TV. This is unusual. Who is right? That's a good question. We only looked at 2 random samples ­ to few to be sure that we are representing the population. We would need to look at a lot more samples and their results after analysis to determine if 75% is too low. HW: p352 C&V #1 ­ 3; p352 #1 ­ 11odd, 15a ­ c, 21, 27, 31 ­ 39 odd
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