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The Central Limit Theorem

Section 7.2 The Central Limit Theorem

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Focus Points For a normal distribution, use and to construct the theoretical sampling distribution for the statistic For large samples use sample estimates to construct a good approximate sampling distribution for the statistic Learn the statement and underlying meaning of the central limit theorem well enough to explain it to a friend who is intelligent, but (unfortunately) doesn’t know much about statistics.

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The Distribution, Given x is normal

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The x Distribution, Given x is normal.

In Section 7.1, we began a study of the distribution of x values, where x was the (sample) mean length of five trout caught by children at the Pinedale children’s fishing pond. Let’s consider this example again in the light of a very important theorem of mathematical statistics.

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The Central Limit Theorem

THEOREM 7.1 The Central Limit Theorem For a normal probability distribution Let x be a random variable with a normal distribution whose mean μ and whose standard deviation is σ. Let be the sample mean corresponding to random samples of size n taken from the x distribution. Then the following are true: a) The distribution is a normal distribution. b) The mean of the distribution is μ. c) The standard deviation of the distribution is .

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The Central Limit Theorem

Theorem 7.1 Explanation: When x has a normal distribution, the distribution will be normal for any sample size n. Furthermore, we can convert the distribution to the standard normal z distribution using the following formulas:

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Example 2(a) – Probability regarding x and x

Suppose a team of biologists has been studying the Pinedale children’s fishing pond. Let x represent the length of a single trout taken at random from the pond. This group of biologists has determined that x has a normal distribution with mean = 10.2 inches and standard deviation = 1.4 inches. What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long?

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Example 2(a) – Solution We use the methods of Section 6.3, with = 10.2 and = 1.4 ,to get

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Example 2(a) – Solution Therefore,

cont’d Therefore, Therefore, the probability is about that a single trout taken at random is between 8 and 12 inches long.

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Example 2(b) – Probability regarding x and x

What is the probability that the mean x length of five trout taken at random is between 8 and 12 inches? Solution: If we let represent the mean of the distribution, then Theorem 6.1, part (b), tells us that If represents the standard deviation of the distribution, then Theorem 6.1, part (c), tells us that

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Example 2 – Solution cont’d To create a standard z variable from , we subtract and divide by : To standardize the interval 8 < < 12, we use 8 and then 12 in place of in the preceding formula for z. 8 < < 12 –3.49 < z < 2.86

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Example 2 – Solution cont’d Theorem 6.1, part (a), tells us that has a normal distribution. Therefore, P(8 < < 12) = P(–3.49 < z < 2.86) = – = The probability is about that the mean length based on a sample size of 5 is between 8 and 12 inches.

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Example 2(c) – Probability regarding x and x

Looking at the results of parts (a) and (b), we see that the probabilities ( and ) are quite different. Why is this the case? Solution: According to Theorem 7.1, both x and have a normal distribution, and both have the same mean of 10.2 inches. The difference is in the standard deviations for x and . The standard deviation of the x distribution is = 1.4.

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Example 2 – Solution The standard deviation of the distribution is

cont’d The standard deviation of the distribution is The standard deviation of the distribution is less than half the standard deviation of the x distribution.

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Example 2 – Solution Figure 7-2 shows the distributions of x and .

cont’d Figure 7-2 shows the distributions of x and . The x distribution with = 10.2 and = 1.4 (b) The x distribution with = 10.2 and = 0.63 for samples of size n = 5

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The x Distribution, Given x is normal.

Looking at Figure 7-2(a) and (b), we see that both curves use the same scale on the horizontal axis. The means are the same, and the shaded area is above the interval from 8 to 12 on each graph. It becomes clear that the smaller standard deviation of the distribution has the effect of gathering together much more of the total probability into the region over its mean. Therefore, the region from 8 to 12 has a much higher probability for the distribution.

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The x Distribution, Given x Follows

any Distribution

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CLT for any distribution

THEOREM 7.2 The Central Limit Theorem for any probability distribution If x possesses any distribution with mean μ and standard deviation is σ, then the sample mean based on a random sample of size n will have a distribution that approaches the distribution of a normal random variable with mean μ and standard deviation as n increase without limit. Question: How large should the sample size be if we want to apply the central limit theorem?

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CLT for any distribution

Answer: After a great deal of theoretical as well as empirical study, statisticians agree that is n is 30 or larger, the distribution will appear to be normal and the central limit theorem will apply. In practice, it is a good idea to make a histogram of sample x values. If the histogram is approximately mound-shaped, and if it is more or less symmetrical, then we may be assured that, the distribution will be well approximated by a normal distribution and the central limit theorem will apply when the sample size n ≥ 30. The main thing to remember is that, a sample size n ≥ 30 is adequate for the central limit theorem to hold. However, in a few rare applications, you may need a sample size larger than 30 to get reliable results.

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The x Distribution, Given x Follows any Distribution

Summary:

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Guided Exercise 2 Suppose x has a normal distribution with mean μ = 18 and standard deviation σ = 3. If you draw random samples of size 5 from the x distribution and represents the sample mean, what can you say about the istribution? = 18, ≈ 1.3

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Guided Exercise 2 (b) Suppose you know that the x distribution has mean μ = 75 and standard deviation σ = 12, but you have no information as to whether or not the x distribution is normal. If you draw samples of size 30 from the x distribution and represents the sample mean, what can you say about the distribution? n = 30 is large enough, the distribution is approximately normal = ≈ 2.2

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Guided Exercise 2 (c) Suppose you did not know that x had a normal distribution. Would you be justified in saying that the distribution is approximately normal if the sample size were n = 8? No, the sample size should be 30 or larger if we don’s know that x has a normal distribution.

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Example 3 – Central Limit Theorem

A certain strain of bacteria occurs in all raw milk. Let x be the bacteria count per milliliter of milk. The health department has found that if the milk is not contaminated, then x has a distribution that is more or less mound-shaped and symmetrical. The mean of the x distribution is μ = 2500, and the standard deviation is σ = 300. In a large commercial dairy, the health inspector takes 42 random samples of the milk produced each day. At the end of the day, the bacteria count in each of the 42 samples is averaged to obtain the sample mean bacteria count .

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Example 3(a) – Central Limit Theorem

Assuming the milk is not contaminated, what is the distribution of ? Solution: The sample size is n = 42. Since this value exceeds 30, the central limit theorem applies, and we know that will be approximately normal, with mean and standard deviation

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Example 3(b) – Central Limit Theorem

Assuming the milk is not contaminated, what is the probability that the average bacteria count for one day is between 2350 and 2650 bacteria per milliliter? Solution: We convert the interval 2650 to a corresponding interval on the standard z axis.

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Example 3(b) – Solution Therefore,

cont’d Therefore, P(2350 2650) = P(–3.24 z 3.24) = – = The probability is that is between 2350 and 2650.

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Example 3(c) – Central Limit Theorem

Interpretation At the end of each day, the inspector must decide to accept or reject the accumulated milk that has been held in cold storage awaiting shipment. Suppose the 42 samples taken by the inspector have a mean bacteria count that is not between 2350 and 2650. If you were the inspector, what would be your comment on this situation?

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Example 3(c) – Solution cont’d The probability that is between 2350 and 2650 for milk that is not contaminated is very high. If the inspector finds that the average bacteria count for the 42 samples is not between 2350 and 2650, then it is reasonable to conclude that there is something wrong with the milk. If is less than 2350, you might suspect someone added chemicals to the milk to artificially reduce the bacteria count. If is above 2650, you might suspect some other kind of biologic contamination.

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The x Distribution, Given x Follows any Distribution

Procedure:

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Guided Exercise 3 In mountain country, major highways sometimes use tunnels instead of long, winding roads over high passes. However, too many vehicles in a tunnel at the same time can cause a hazardous situation. Traffic engineers are studying a long tunnel in Colorado. If x represents the time for a vehicle to go through the tunnel, it is known that the x distribution has mean μ = 12.1 minutes and standard deviation σ = 3.8 minutes under ordinary traffic conditions. From a histogram of x values, it was found that the x distribution is mound-shaped with some symmetry about the mean.

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Guided Exercise 3 Engineers have calculated that, on average, vehicles should spend from 11 to 13 minutes in the tunnel. If the time is less than 11 minutes, traffic is moving too fast for safe travel in the tunnel. If the time is more than 13 minutes, there is a problem of bad air quality (too much carbon monoxide and other pollutants). Under ordinary conditions, there are about 50 vehicles in the tunnel at one time. What is the probability that the mean time for 50 vehicles in the tunnel will be from 11 to 13 minutes? We will answer this question in steps.

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Guided Exercise 3 (a) Let represent the sample mean based on samples of size 50. Describe the distribution. Approximately normal, with = μ = 12.1 = ≈ 0.54 (b) Find P(11 < < 13) (c) Interpretation your answer to part (b) 93% of the time, there should be no safety hazard for average traffic flow.

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