The Central Limit Theorem says that the standard deviation of the sampling distribution is
Sampling Distribution of the Sample MeansInstead of working with individual scores, statisticians often work with means. What happens is that several samples are taken, the mean is computed for each sample, and then the means are used as the data, rather than individual scores being used. The sample is a sampling distribution of the sample means. Show
When all of the possible sample means are computed, then the following properties are true:
The formula for a z-score when working with the sample means is: Finite Population Correction FactorIf the sample size is more than 5% of the population size and the sampling is done without replacement, then a correction needs to be made to the standard error of the means. In the following, N is the population size and n is the sample size. The adjustment is to multiply the standard error by the square root of the quotient of the difference between the population and sample sizes and one less than the population size. For the most part, we will be ignoring this in class. Table of Contents Central limit theorem less than example 2 Can’t see the video? Click here. The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger — no matter what the shape of the population distribution. This fact holds especially true for sample sizes over 30. All this is saying is that as you take more samples, especially large ones, your graph of the sample means will look more like a normal distribution. Here’s what the Central Limit Theorem is saying, graphically. The picture below shows one of the simplest types of test: rolling a fair die. The more times you roll the die, the more likely the shape of the distribution of the means tends to look like a normal distribution graph. Photo credit:cmglee|Wikimedia CommonsThe Central Limit Theorem and MeansAn essential component of the Central Limit Theorem is that the average of your sample means will be the population mean. In other words, add up the means from all of your samples, find the average and that average will be your actual population mean. Similarly, if you find the average of all of the standard deviations in your sample, you’ll find the actual standard deviation for your population. It’s a pretty useful phenomenon that can help accurately predict characteristics of a population. Watch a video explaining this phenomenon, or read more about it here: The Mean of the Sampling Distribution of the Mean. Definition using Calculus
Note: An assumption is that the expected value of X and X2 < infinity.
A Central Limit Theorem word problem will most likely contain the phrase “assume the variable is normally distributed”, or one like it. With these central limit theorem examples, you will be given:
Click the link to skip down to one of three central limit theorem examples:
Central Limit Theorem Examples: Greater thanFor Central Limit Theorem word problems that contain the phrase “greater than” (or a similar phrase such as “above”). Watch the video for an example. Central Limit Theorem Greater Than Example #3 Can’t see the video? Click here. 1. General Steps
Step 2: Draw a graph. Label the center with the mean. Shade the area roughly above (i.e. the “greater than” area). This step is optional, but it may help you see what you are looking for. Step 3: Use the following formula to find the z-score. Plug in the numbers from step 1. Click here if you want easy, step-by-step instructions for solving this formula.
Step 4: Look up the z-score you calculated in step 3 in the z-table. If you don’t remember how to look up z-scores, you can find an explanation in step 1 of this article: Area to the right of a z-score. Step 5: Subtract your z-score from 0.5. For example, if your score is 0.1554, then 0.5 – 0.1554 = 0.3446. Step 6: Convert the decimal in Step 5 to a percentage. In our example, 0.3446 = 34.46%. That’s it! 2. Specific Example Q. A certain group of welfare recipients receives SNAP benefits of $110 per week with a standard deviation of $20. If a random sample of 25 people is taken, what is the probability their mean benefit will be greater than $120 per week? Step 1: Insert the information into the z-formula: Back to top for more Central Limit Theorem Examples Central Limit Theorem Examples: Less thanSolving Central Limit Theorem word problems that contain the phrase “less than” (or a
similar phrase such as “lower”). Central limit theorem less than example 2 Can’t see the video? Click here. 1. General Steps
Step 2: Draw a graph. Label the center with the mean. Shade the area roughly below (i.e. the “less than” area). This step is optional, but it may help you see what you are looking for. Step 3: Use the following formula to find the z-score. Plug in the numbers from step 1. Click here if you want simple, step-by-step instructions for using this formula.
Step 4: Look up the z-score you calculated in step 4 in the z-table. If you don’t remember how to look up z-scores you can find an explanation in step 1 of this article on area to the right of a z-score in a normal distribution curve. Step 5: Add your z-score to 0.5. For example, if your z-score is 0.1554, then 0.5 + 0.1554 is 0.6554. Step 6: Convert the decimal in Step 6 to a percentage. In our example, 0.6554 = 65.54%. That’s it! 2. Specific Example Step 1: Insert the values into the z-formula: Step 2: Look up the z-score in the left-hand z-table (or use technology). -1.51 has an area of 93.45%. Back to top for more Central Limit Theorem Examples Central Limit Theorem Examples: BetweenWatch the video for an example:
Central Limit Theorem Example Can’t see the video? Click here. Example problem: There are 250 dogs at a dog show who weigh an average of 12 pounds, with a standard deviation of 8 pounds. If 4 dogs are chosen
at random, what is the probability they have an average weight of greater than 8 pounds and less than 25 pounds? Step 1:Identify the parts of the problem. Your question should state:
Step 2:
Draw a graph. Label the center with the mean. Shade the area between 1 and 2. This step is optional, but it may help you see what you are looking for. Step 3: Use the following formula to find the z-scores. All this formula is asking you to do is: a) Subtract the mean (μ in Step 1) from the greater than value (Xbar in Step 1): 25 – 12 = 13. Step 4 Use the formula from Step 3 to find the z-values. This time, use Xbar2 from Step 1 (8). a) Subtract the
mean (μ in Step 1) from the greater than value (Xbar in Step 1): 8 – 12 = -4. Step 5: Look up the value you calculated in Step 3 in the z-table. Z value of 3.25 corresponds to .4994 Step 6: Look up the value you calculated in Step 4 in the z-table. Z value of 1 corresponds to .3413 Note that the bell curve is symmetrical, so if you want to look up a negative value like -1, then just look up the positive counterpart. The area will be the same. Step 7: Add Step 5 and 6 together: .4994 + .3413 = .8407 Step 8: Convert the decimal in Step 7 to a percentage: .8407 = 84.07% That’s it! Central Limit Theorem on the TI 89Example problem: A population of community college students includes inner city students (p = .33). What is the probability that a random sample of 45 students from the population will have from 20% to 40% inner city students? Step 1: Press APPS. Highlight the Stats/List Editor by using the scroll keys. Press ENTER. Step 2: Press F5 and scroll down to C: BinomialCdf. Step 3: Enter 45 in the Num Trials box. Step 4: Scroll down and enter .33 in the Prob Success box. Step 5: Scroll down and enter 9 in the Lower Value box (because 20% of 45 = 9). Step 6: Scroll down and enter 18 in the Upper Value box (because 40% of 45 = 18). Press ENTER. Step 7: Read the Result: Cdf = .857142. This means that the probability your random sample will have 20 – 40% inner city students is 85.71%. TI 83 Central Limit Theorem: OverviewWatch the video or read the article below:
TI 83 Central Limit Theorem Examples Can’t see the video? Click here. The TI 83 calculator has a built in function that can help you calculate probabilities of central theorem word problems, which usually contain the phrase “assume the distribution is normal” (or a variation of that phrase). The function, normalcdf, requires you to enter a lower bound, upper bound, mean, and standard deviation. Example problem: A fertilizer company manufactures organic fertilizer in 10 pound bags with a standard deviation of 1.25 pounds per bag. What is the probability that a random sample of 15 bags will have a mean between 9 and 9.5 pounds? Step 1: 2nd VARS 2. Step 2: Enter your variables (lower bound, upper bound, mean, and standard deviation). Separate each variable by a comma: 9,9.5, 10,(1.25/√15)). Step 3: Press ENTER. This returns the probability of .05969, or .05969%. Tip:If you have a question that asks for “greater than” or “less than” a certain number, enter 999999999 for the lower or upper bound. For
example, if you wanted to know the probability of greater than 8 pounds you would enter: Tip: Sampling distributions require that the standard deviation of the mean is σ / √(n), so make sure you enter that as the standard deviation. Check out our statistics YouTube channel for more tips and central limit theorem examples! ReferencesGonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial. ---------------------------------------------------------------------------
Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free! Comments? Need to post a correction? Please Contact Us. What does the central limit theorem says about sampling distribution?The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal.
What standard deviation of the sampling distribution of the sample means according to the central limit theorem?According to Central Limit Theorem, if the sample size is larger, the sampling distribution of the means becomes closer to normal. The standard deviation of the sampling distribution of the means will decrease and it will be almost the same as the standard deviation of X as the sample size increases.
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