Because our sample size was at least 10 well, barely! Then, the two conditions are met if:. Does that mean all of our discussion here is for naught? No, not at all! In Stat , we'll use the sample proportion in conjunction with the above result to draw conclusions about the unknown population proportion p. You'll definitely be seeing much more of this in Stat ! Breadcrumb Home 28 Font size. Font family A A. We do this by converting the range of values into standardized units and finding the area under the normal curve.
In other words, the scope of the normal approximation is dependent upon our sample size, becoming more accurate as the sample size grows. This characteristic follows with the statistical themes of the law of large numbers and central limit theorem reviewed below. Law of Large Numbers : An illustration of the law of large numbers using a particular run of rolls of a single die.
As the number of rolls in this run increases, the average of the values of all the results approaches 3. While different runs would show a different shape over a small number of throws at the left , over a large number of rolls to the right they would be extremely similar.
The law of large numbers LLN is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins.
Any winning streak by a player will eventually be overcome by the parameters of the game. It is important to remember that the LLN only applies as the name indicates when a large number of observations are considered. The central limit theorem CLT states that, given certain conditions, the mean of a sufficiently large number of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed.
The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions, given that they comply with certain conditions. In this atom, we provide an example on how to compute a normal approximation for a binomial distribution.
The following is an example on how to compute a normal approximation for a binomial distribution. Assume you have a fair coin and wish to know the probability that you would get 8 heads out of 10 flips. The standard deviation is, therefore, 1.
A total of 8 heads is:. Standard deviations above the mean of the distribution. The problem is that the binomial distribution is a discrete probablility distribution whereas the normal distribultion is a continuous distribution. The solution is to round off and consider any value from 7. Sign up Log in. Poisson Approximation The normal distribution can also be used to approximate the Poisson distribution for large values of l the mean of the Poisson distribution. Continuity Correction The binomial and Poisson distributions are discrete random variables, whereas the normal distribution is continuous.
In the discrete distribution, each probability is represented by a rectangle right hand diagram : When working out probabilities, we want to include whole rectangles, which is what continuity correction is all about. Example Suppose we toss a fair coin 20 times. You must meet the conditions for a binomial distribution:. The shape of the binomial distribution needs to be similar to the shape of the normal distribution. In order to get the best approximation, add 0. The number 0. Suppose in a local Kindergarten through 12 th grade K - 12 school district, 53 percent of the population favor a charter school for grades K through 5.
A simple random sample of is surveyed. The mean is and the standard deviation is 8.
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