Examples of real number in the following topics:
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- X = a real number between a and b (in some instances, X can take on the values a and b). a = smallest X ; b = largest X
- X = a real number, 0 or larger. m = the parameter that controls the rate of decay or decline
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- A variable is any characteristic, number, or quantity that can be measured or counted.
- A variable is any characteristic, number, or quantity that can be measured or counted.
- Numeric variables have values that describe a measurable quantity as a number, like "how many" or "how much. " Therefore, numeric variables are quantitative variables.
- Observations can take any value between a certain set of real numbers.
- Examples of discrete variables include the number of registered cars, number of business locations, and number of children in a family, all of of which measured as whole units (i.e., 1, 2, 3 cars).
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- The following data are real.
- The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as follows: (Source: HIV/AIDS Epidemiology Santa Clara County, Santa Clara County Public Health Department, May 2011)
- If the ethnicity of AIDS victims followed the ethnicity of the total county population, fill in the expected number of cases per ethnic group.
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- A fair die has an equal probability of landing face-up on each number.
- A die (plural dice) is a small throw-able object with multiple resting positions, used for generating random numbers.
- An example of a traditional die is a rounded cube, with each of its six faces showing a different number of dots (pips) from one to six.
- Thus, they are a type of hardware random number generator.
- All such dice are stamped with a serial number to prevent potential cheaters from substituting a die.
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- Imagine rolling a large number of identical, unbiased dice.
- Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal probability distribution.
- The central limit theorem has a number of variants.
- $n$ is the number of values that are averaged together not the number of times the experiment is done.
- The sample means are generated using a random number generator, which draws numbers between 1 and 100 from a uniform probability distribution.
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- The easiest way to understand the mean, variance, and standard deviation of the binomial distribution is to use a real life example.
- Consider a coin-tossing experiment in which you tossed a coin 12 times and recorded the number of heads.
- If you performed this experiment over and over again, what would the mean number of heads be?
- Therefore, the mean number of heads would be 6.
- In general, the mean of a binomial distribution with parameters $N$ (the number of trials) and $p$ (the probability of success for each trial) is:
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- While there is a real theorem that a random variable will reflect its underlying probability over a very large sample (the law of large numbers), the law of averages typically assumes that unnatural short-term "balance" must occur.
- In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times.
- It is important to remember that the law of large numbers only applies (as the name indicates) when a large number of observations are considered.
- 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.
- Evaluate the law of averages and distinguish it from the law of large numbers.
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- Many distributions in real life can be approximated using normal distribution.
- In a probability histogram, the height of each bar shows the true probability of each outcome if there were to be a very large number of trials (not the actual relative frequencies determined by actually conducting an experiment).
- We study the normal distribution extensively because many things in real life closely approximate the normal distribution, including:
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- The following real data are the result of a random survey of 39 national flags (with replacement between picks) from various countries.
- We are interested in finding a confidence interval for the true mean number of colors on a national flag.
- Let X = the number of colors on a national flag.
- Construct a 95% Confidence Interval for the true mean number of colors on national flags.
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- The normal (Gaussian) distribution is a commonly used distribution that can be used to display the data in many real life scenarios.
- Normal distributions are extremely important in statistics, and are often used in the natural and social sciences for real-valued random variables whose distributions are not known.
- One reason for their popularity is the central limit theorem, which states that, under mild conditions, the mean of a large number of random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution.
- Another reason is that a large number of results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically, in explicit form, when the relevant variables are normally distributed.
- The normal distribution is symmetric about its mean, and is non-zero over the entire real line.