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MTHE 224

Week 4: Normal Distributions and Confidence Intervals Goals: • Study normal and standard normal distributions • Introduce confidence intervals • Study confidence intervals based on normal distributions Suggested Textbook Readings: Chapter 4: 4.3, 4.6 Chapter 7: 7.1 Practice Problems: Section 4.3: 53, 55 Section 4.6: 89 Section 7.1: 1, 3, 5 Week 4: Normal Distributions and Confidence Intervals 2 Normal approximation to the binomial distribution It turns out that when n is large, the binomial distribution starts to look like a normal distribution. Recall that the binomial distribution is the number of successes out of n trials of an experiment, where the success probability is p. We found that the expected value of a binomial random variable X ∼ Bin(n,p)w s E[X]= np and the variance was V [X]= np(1 − p). If you have a large number of samples, you can approximate the (discrete) binomial distribution by the (continuous) normal distribution. Before you do this, you must check that n is large enough. If np,n(1 − p) ≥ 10, n is considered large enough. Proposition: If X ∼ Bin(n,p) with np ≥ 10 and n(1 − p) ≥ 10, then a normal approximation to the binomial distribution may be used. That is, X has an approximate N(np, ▯ np(1 − p)) distribution and P(X ≤ x)= B(x;n,p) ≈ ( Area under the normal curve to the left of (x +0 .5)) ▯ ▯ =Φ ▯ +0 .5 − np . np(1 − p) Example 1: Suppose Princess street has 50 parking spots on it. A parking spot is empty with probability 0.2. Use the normal approximation to the binomial distribution to find the following probabilities (a) What is the (approximate) probability that there are 4 or fewer empty parking spots on all of Princess st? (b) What is the (≈) probability that there are between 6 and 12 free spots? (c) What are the exact probabilities? What is the error in using the normal approxima- tion? MTHE 224 Fall 2012 Week 4: Normal Distributions and Confidence Intervals 3 Probability Plot In many instances, we need to check whether a sample data set appears to be generated by a normally distributed variable. The probability plot is an effective tool to detect departures from normality. Example 2: Suppose we have a set of four observations: 6, 4, 7, 8. Determine if the set come from a normal distribution. observed values ✻ 8 6 4 z normal score ✲ (z percentiles) -1.2 -0.8 -0.4 0 0.4 0.8 1.2 Definition: Normal scores, for some sample size n, refer to an idealized standard normal distribution. It consists of the values of z that divide that axes into n equal probability intervals. The ith normal score is taken to be the [100(i − 0.5)/n]th percentile, for i =1 ,2,··· ,n. To construct a probability plot for a sample size n, (1) order the data from small to largest, (2) obtain the normal scores for all i, (3) pairing the the ordered observations and the normal scores, for all i, (4) make a scattered plot of the resulting points.
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