Question

Page < 6 > of 12 Lesson 6.2.4: Binomial Distribution and Sample Proportions NEXT STEPS We have formulas for the mean (center) and standard deviation (spread) of a distribution of sample proportions. Don't forget the shape! Your graph was bell shaped-symmetric, high in the middle and low at the ends. It is similar to a normal distribution, not smooth, but still bell shaped. Your instructor will now use a computer applet to demonstrate the way that binomial distributions and distributions of sample proportions vary with changing sample size and probability of success. Pay attention to the changes in the shape of the distribution as the sample size (n) and probability of success (p) change. Your job is to learn when a binomial distribution is approximately normal, depending on the values of n and p. 14 When p = 0.5, what is a common property of the shapes for all sample sizes n? What change do you notice as n increases? 15 When p = 0.25, what is a common property of the shapes for all sample sizes (n)? What change do you notice as n increases? 16 When p = 0.80, what is a common property of the shapes for all sample sizes (n)? What change do you notice as n increases? We have seen that when we increase the value of n sufficiently, the binomial distribution becomes symmetric and bell-shaped like the normal distribution. As long as the distribution has a left tail (whose length is the mean number of successes, np) that is long enough, and a right tail (whose length is the mean number of failures, n(1-P)) that is long enough, the distribution will be approximately normal. Most statisticians agree that the left and right tails should be at least 10 in length. MacBook Pro

Answer 1

14. When in a Binomial Distribution, p = 0.5 and n is sufficiently large, it will follow a bell shaped curve like normal with mean = np and variance = np(1-p). In general np should be >= 10

15. When in a Binomial Distribution, p = 0.25, then we should have np >=10 to have a normal distribution approximation. Therefore the value of n should be >= 40, to have a normal approximation with mean = np and variance = np(1-p)

16. When in a Binomial Distribution, p = 0.80, then we should have np >=10 to have a normal distribution approximation. Therefore the value of n should be >= 13, to have a normal approximation with mean = np and variance = np(1-p)

17. Normal distribution is a good approximation of normal distribution when p is near the value of 0.5 and n is sufficiently large to have np >= 10

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