The Central Limit Theorem states that even if the population is not normally distributed the Quizlet

1. According to Central Limit theorem, which sample size will give a smaller standard error of the mean? A. 7 B. 12 C. 23 D. 40 2. If a population is not normally distributed, the distribution of the sample means for a given sample size n will ____________. A. be positively skewed. B. be negatively skewed. C. take the same shape as the population. D. approach a normal distribution as n increases. 3. The mean and standard deviation of a population are 75 and 15, respectively. The sample size is 100. What is the standard error of the mean? A. 1.5 B. 1.73 C. 0.15 D. 8 4. The mean and standard deviation of a population are 400 and 40, respectively. Sample size is 25. What is the mean of the sampling distribution? A. 400 B. 40 C. 25 D. 8 5. What is the standard error of the mean if the sample size is 25 with standard deviation of 16? A. 6.25 B. 3.2 C. 1.25 D. 0.64 6. The weights of the eggs produced by a certain breed of hen are normally distributed with mean 65 grams and standard deviation of 5 grams. Which of the following will you use? A. Normal Distribution C. Discrete Probability Distribution B. Central Limit Theorem D. Binomial Distribution 7. In a study done on the life expectancy of 500 people in a certain geographic region, the mean age at death was 72 years and the standard deviation was 5.3 years. If a sample of 50 people from this region is selected, and the probability that the mean life expectancy will be less than 70 years, which of the following will you use? A. Normal Distribution C. Discrete Probability Distribution B. Central Limit Theorem D. Binomial Distribution 16 8. The mean and standard deviation of a population are 200 and 20, respectively. What is the probability of selecting 25 data values with a mean less than 190? A. 69% B. 31% C. 0.6% D. 99% 9. In a metal fabrication process, metal rods are produced that have an average length of 20.5 meters with a standard deviation of 2.3 meters. A quality control specialist collects a random sample of 30 rods and measures their lengths. Suppose the resulting sample mean is 19.5 meters. Which of the following statements is true? A. This sample mean is 2.38 standard deviations above what we expect. B. This sample mean is 2.38 standard deviations below what we expect. C. This sample mean is only 1 standard deviation above the population mean. D. This sample mean is more than 3 standard deviations away from the population mean. For number 10-11, refer to the problem below. Suppose the teenagers that attend public high schools get an average of 5.7 hours of sleep each night with a standard deviation of 1.7 hours. Assume that the average sleep hour is normally distributed, and 35 high school students are randomly selected. 10.Compute the z-score for 6 hours of sleep. A. 1.04 B. 0.18 C. 0.52 D. 0.82 11.What is the probability that a randomly selected group of 35 high school students gets more than 6 hours of sleep each night? A. 0.3508 B. 0.1492 C. 0.0714 D. 0.4286 For number 12-14, refer to the problem below. The amount of fuel used by jumbo jets to take off is normally distributed with a mean of 4, 000 gallons and a standard deviation of 125 gallons. A sample of 40 jumbo jets are randomly selected. 12.Compute the z-score for 3, 950 gallons. A. – 0.4 B. 0.4 C. – 2.53 D. 2.53 13.What is the probability that the mean number of gallons of fuel needed to take off for a randomly selected sample of 40 jumbo jets will be less than 3, 950 gallons? a. 78.1% B. 34.5% C. 2.5% D. 0.57% 14.What is the probability that the mean number of gallons of fuel needed to take off for a randomly selected sample of 40 jumbo jets will be more than 3, 950 gallons? b. 0.57% B. 49.43% C. 65.54% D. 99.43% 15.Researchers found that boys playing high school football recorded an average of 355 hits to the head with a standard deviation of 80 hits during a season. What is the probability on a randomly selected team of 48 players that the average number of head hits per player is between 340 and 360? A. 56.96% B. 43.04% C. 40.32% D. 16.64%

Central Limit Theorem

The central limit theorem states that the sampling distribution of the mean of any independent,random variable will be normal or nearly normal, if the sample size is large enough.

How large is "large enough"? The answer depends on two factors.

• The shape of the underlying population. The more closely the original population resembles a normal distribution, the fewer sample points will be required.

In practice, some statisticians say that a sample size of 30 is large enough when the population distribution is roughly bell-shaped. Others recommend a sample size of at least 40. But if the original population is distinctly not normal (e.g., is badly skewed, has multiple peaks, and/or has outliers), researchers like the sample size to be even larger.

T-Distribution vs. Normal Distribution

The t distribution and the normal distribution can both be used with statistics that have a bell-shaped distribution. This suggests that we might use either the t-distribution or the normal distribution to analyze sampling distributions. Which should we choose?

Guidelines exist to help you make that choice. Some focus on the population standard deviation.

• If the population standard deviation is unknown, use the t-distribution.

Other guidelines focus on sample size.

• If the sample size is small, use the t-distribution.

In practice, researchers employ a mix of the above guidelines. On this site, we use the normal distribution when the population standard deviation is known and the sample size is large. We might use either distribution when standard deviation is unknown and the sample size is very large. We use the t-distribution when the sample size is small, unless the underlying distribution is not normal. The t distribution should not be used with small samples from populations that are not approximately normal.

Test Your Understanding

In this section, we offer two examples that illustrate how sampling distributions are used to solve commom statistical problems. In each of these problems, the population sample size is known; and the sample size is large. So you should use the Normal Distribution Calculator, rather than the t-Distribution Calculator, to compute probabilities for these problems.

Normal Distribution Calculator

The normal calculator solves common statistical problems, based on the normal distribution. The calculator computes cumulative probabilities, based on three simple inputs. Simple instructions guide you to an accurate solution, quickly and easily. If anything is unclear, frequently-asked questions and sample problems provide straightforward explanations. The calculator is free. It can be found under the Stat Tables tab, which appears in the header of every Stat Trek web page.

Example 1

Assume that a school district has 10,000 6th graders. In this district, the average weight of a 6th grader is 80 pounds, with a standard deviation of 20 pounds. Suppose you draw a random sample of 50 students. What is the probability that the average weight of a sampled student will be less than 75 pounds?

Solution: To solve this problem, we need to define the sampling distribution of the mean. Because our sample size is greater than 30, the Central Limit Theorem tells us that the sampling distribution will approximate a normal distribution.

To define our normal distribution, we need to know both the mean of the sampling distribution and the standard deviation. Finding the mean of the sampling distribution is easy, since it is equal to the mean of the population. Thus, the mean of the sampling distribution is equal to 80.

The standard deviation of the sampling distribution can be computed using the following formula.

σx = [ σ / sqrt(n) ] * sqrt[ (N - n ) / (N - 1) ] 
σx = [ 20 / sqrt(50) ] * sqrt[ (10,000 - 50 ) / (10,000 - 1) ] = (20/7.071) * (0.995) = 2.81

Let's review what we know and what we want to know. We know that the sampling distribution of the mean is normally distributed with a mean of 80 and a standard deviation of 2.82. We want to know the probability that a sample mean is less than or equal to 75 pounds.

Because we know the population standard deviation and the sample size is large, we'll use the normal distribution to find probability. To solve the problem, we plug these inputs into the Normal Probability Calculator: mean = 80, standard deviation = 2.81, and normal random variable = 75. The Calculator tells us that the probability that the average weight of a sampled student is less than 75 pounds is equal to 0.038.

Note: Since the population size is more than 20 times greater than the sample size, we could have used the "approximate" formula σx = [ σ / sqrt(n) ] to compute the standard error. Had we done that, we would have found a standard error equal to [ 20 / sqrt(50) ] or 2.83.

Example 2

Find the probability that of the next 120 births, no more than 40% will be boys. Assume equal probabilities for the births of boys and girls. Assume also that the number of births in the population (N) is very large, essentially infinite.

Solution: The Central Limit Theorem tells us that the proportion of boys in 120 births will be approximately normally distributed.

The mean of the sampling distribution will be equal to the mean of the population distribution. In the population, half of the births result in boys; and half, in girls. Therefore, the probability of boy births in the population is 0.50. Thus, the mean proportion in the sampling distribution should also be 0.50.

The standard deviation of the sampling distribution (i.e., the standard error) can be computed using the following formula.

σp = sqrt[ PQ/n ] * sqrt[ (N - n ) / (N - 1) ]

Here, the finite population correction is equal to 1.0, since the population size (N) was assumed to be infinite. Therefore, standard error formula reduces to:

σp = sqrt[ PQ/n ] 
σp = sqrt[ (0.5)(0.5)/120 ] = sqrt[0.25/120 ] = 0.04564

Let's review what we know and what we want to know. We know that the sampling distribution of the proportion is normally distributed with a mean of 0.50 and a standard deviation of 0.04564. We want to know the probability that no more than 40% of the sampled births are boys.

Because we know the population standard deviation and the sample size is large, we'll use the normal distribution to find probability. To solve the problem, we plug these inputs into the Normal Probability Calculator: mean = .5, standard deviation = 0.04564, and the normal random variable = .4. The Calculator tells us that the probability that no more than 40% of the sampled births are boys is equal to 0.014.

Note: This problem can also be treated as a binomial experiment. Elsewhere, we showed how to analyze a binomial experiment. The binomial experiment is actually the more exact analysis. It produces a probability of 0.018 (versus a probability of 0.14 that we found using the normal distribution). Without a computer, the binomial approach is computationally demanding. Therefore, many statistics texts emphasize the approach presented above, which uses the normal distribution to approximate the binomial.

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