The Underlying Distribution of the Underlying Random Variable

Transcription

1 Sampling Theory Sampling Theory The Underlying Distribution of the Underlying Random Variable Let X 1, X 2, ellipsis, X n denote a random sample of size n from some population with population mean mu and with population variance sigma 2. Alternatively, X 1, X 2, ellipsis, X n can be the results of n independent and identical experiments, i.e., X 1, X 2, ellipsis, X n can be the results of n independent replications or trials of the same experiment. In either case, we adopt the following model for how the data X 1, X 2, ellipsis, X n were obtained. The Model: Each X i, i = 1, ellipsis, n, isanindependent observation of a random variable X with mean mu and variance sigma 2. Statistics and Parameters The random variable X is called the underlying random variable, and its (perhaps unknown) distribution is called the underlying distribution. Furthermore, regardless of whether the data came from sampling a population or from replicating an experiment, we take the liberty of referring to the mean mu and the variance sigma 2 of the underlying distribution, as the (underlying) population mean and variance. The primary reason for these names is to distinguish between the parameters mu and sigma of the probability distribution of X, the statistics X and s of the empirical distribution of X, and a third mean and variance/standard deviation, which will be introduced presently. Note: The sample mean and sample variance are statistics that describe the empirical distribution, and thereby the sample, by describing its location and dispersion. The population mean and population variance are parameters that describe the underlying probability distribution, and thereby the population (or experimental phenomena), in the same way: by describing its location and dispersion. The difference is that, according to our model, the parameters are (perhaps unknown and unobservable) constants, whereas the statistics are observable random variables. Statistics are observable random variables that describe (the empirical distribution of) a sample. Statistics are observable in the sense that their observed values depend only on the values of the sample data. Statistics are random in the sense that their observed values depend on chance (the chance involved in drawing the sample). Statistics are variables in the sense that if we were to draw a second sample of the same size and under identical conditions, then the observed values of the statistics of the second sample would be different (vary) from those of the first sample, because of the chance of the draw. October 20, (SAMPLING)

2 G.I. Holtzman, Fall 1997 Parameters are nonrandom constants that describe (the probability distribution of) a population or an experimental phenomenon. Parameters are nonrandom constants: there is only one underlying population (or experimental phenomenon) of interest, and it has only one mean, and only one variance. Moreover, parameters are often unobservable, otherwise we wouldn't be working with samples. The Sampling Distribution of the Mean Focus on the sample mean, X. Like any statistic, it is a random variable. Like any random variable, it has a probability distribution. The probability distribution of a statistic is sometimes called a sampling distribution to distinguish it from the empirical distribution and from the underlying distribution. The probability distribution of X is called the sampling distribution of the mean. Like any probability distribution, a sampling distribution has parameters by which it can be summarized and described. The location of the sampling distribution of X is described by its mean or expected value E(X) == mu X. The dispersion of the sampling distribution of the sample mean is described by its variance, Var(X) == sigma 2 X, or by its square root, sigma X. The parameter sigma X is, of course, the standard deviation of (the sampling distribution of the sample mean) X, but traditionally it is also known as the standard error (of the mean). Recapitulation and Summary All we have done so far is model, name, define, and denote. We now have three distributions with three means and three standard deviations. X and s are statistics of the empirical distribution mu X and sigma X are parameters of the underlying distribution mu X and sigma X are parameters of the sampling distribution of the (sample) mean Sampling Theory Since X is a statistic, i.e., since X == (X 1 + X 2 + ellipsis + X n )/n is a function of the sample alone (and of no unknown parameters), it stands to reason that That is, Specifically, if we knew the (underlying) distribution of X, then we could derive the (sampling) distribution of X. The distribution of X determines the distribution of X. mu X determines mu X (location) sigma X determines sigma X (dispersion) October 20, (SAMPLING)

3 Sampling Theory and the shape of the distribution of X determines the shape of the distribution of X. Theorem 1 (location): If X is the mean of a random sample of size n of any random variable X, then mu X = mu X. Theorem 2 (dispersion): If X is the mean of a random sample of size n of any random variable X, then sigma 2 X = sigma 2 X n i.e., sigma X = sigma X sqrtn. Theorem 3 (shape): Let X be the mean of a random sample of size n of a random variable X. If X is normally distributed, then X is normally distributed. Corollary A (of 1, 2 and 3): If X is the mean of a random sample of size n of a random variable X, and if X similar N(mu, sigma 2 ), then X similar N(mu, sigma 2 /n). Moreover, if Z == X - mu sigma/sqrtn, then Z similar N(0, 1). The Central Limit Theorem (CLT) The Central Limit Theorem (shape): If X is the mean of a random sample of size n of any random variable X, and if n is large, then X is approximately normally distributed. Corollary B (of 1, 2 and the CLT): If X is any random variable with mean mu and variance sigma 2, if X is the mean of a random sample of size n, and if n is large, then X similar N(mu, sigma 2 /n) approximately. Moreover, if Z == X - mu sigma/sqrtn, then Z similar N(0, 1) approximately. Note: The word any as used in the phrase any random variable in Theorems 1 and 2, and in the CLT means any quantitative (real or integer valued) random variable regardless of the functional form or shape of its distribution. In particular, Theorems 1 and 2 are true for non-normally distributed random variables as well as for normally distributed random variables. Exercises Instructions: These exercises should be done on your own inch paper to hand in. Answer with complete sentences, and please do not hand in the handout. X-1. Define a simple random sample. October 20, (SAMPLING)

4 G.I. Holtzman, Fall 1997 X-2. The Normal Family. The yearly growth of dwarf-apple-tree seedlings can be measured by the increase in the length of the central leader. Suppose that the second-year growth of such trees is normally distributed with a mean of 20 cm and a standard deviation of 6 cm. 1. Compute the probability that the second-year growth of a randomly selected two-year-old dwarf-apple-tree seedling is less than 15 cm. 2. Compute the percentage of such dwarf-apple-tree seedlings that would grow more than 25 cm. 3. Compute the fraction of such dwarf-apple-tree seedlings that would be expected to have a second-year growth of between 10 and 30 cm. 4. Find a number x such that the second-year growth of 90% of the seedlings is more than x. 5. Find two numbers a and b such that the second-year growth of 90% of the seedlings is between a and b and such that the second-year growth of 5% is less than a and the second-year growth of 5% is more than b. X-3. Reconsider the dwarf-apple trees of exercise X-2 Let the random variable X represent the second-year growth of a randomly sampled dwarf-apple-tree seedling. Note that the characteristic of interest here is second-year growth, while the (underlying) population of interest is all two-year-old dwarf-apple-tree seedlings. 1. How do we denote the sample of n observations of X drawn from the underlying population? X 1, X 2, ellipsis, X n 2. How do we denote (i.e. give the symbol for) and define (i.e., give the formula for) the mean of those n observations? X = sumx i /n 3. Is that the sample mean or the population mean? sample mean 4. Is it a random variable or a constant? random variable 5. Is it a statistic or a parameter? statistic X-4. Continuing the previous question How do we denote the population mean? &mu. or mu X 2. Is it a random variable or a constant? constant 3. Is it a statistic or a parameter? parameter X-5. Continuing the previous question How do we denote the mean of the sampling distribution of the mean? mu X 2. Is it a random variable or a constant? constant 3. Is it a statistic or a parameter? parameter 4. To which am I referring in 1., the sampling distribution of the sample mean, or of the population mean? sample 5. What is the relationship between the mean of the sampling distribution of the mean and the mean of the underlying population? (Write the equation.) mu X = mu or mu X = mu X 6. Which theorem gives that relationship? Theroem 1 7. What statistic do we use to estimate the population mean? X October 20, (SAMPLING)

5 Sampling Theory X-6. Continuing the previous question How do we denote the standard deviation of the sampling distribution of the mean? sigma X 2. What is the special name for this standard deviation? standard error of the mean 3. How do we denote the standard deviation of the underlying population? &sigma. or sigma X 4. How do we denote the standard deviation of the sample? s 5. Which of these three are parameters? sigma X and sigma or sigma X 6. Which of these three are statistics? s 7. Which of these three are constants? sigma X and sigma or sigma X 8. Which of these three are random variables? s 9. Write down the equation that relates the standard error of the mean to the standard deviation of the underlying population. sigma X = sigma/sqrtn 10. Which theorem gives that relationship? Theorem What statistic do we use to estimate the standard error of the mean? (give its formula as a function of the sample standard deviation and the sample size). s/sqrtn X-7. Sample Size. 1. Again, write down the equation that relates the standard error of the mean to the standard deviation of the underlying population. sigma X = sigma/sqrtn 2. Recall that the standard deviation of the underlying population of apple-trees in exercise X-62 on page 58 is sigma = 6 cm. Tabulate the values of the standard error corresponding to samples of size 1, 4, 9, 16, 25, and Graph the relationship with the sample size n on the horizontal axis and the standard error sigma/sqrtn on the vertical axis. 4. As the sample size increases, does the standard error of the mean increase or decrease? decrease 5. As the sample size increases, does the dispersion of the sample mean around the population mean increase or decrease? decrease 6. Does a larger sample give you more, or less, information about the population (mean)? more 7. With sigma = 6, how large would n have to be to to reduce the standard error to 0.5? X-8. Again referring to exercise X-72 on page 64 and to the notation developed for the second-year growth X, compute the following. 1. Compute P»15 < X < 25{ P»15 < X < 25{ = P» -.83 < Z <.83{ = For samples of size n = 9, compute P»15 < X < 25{. (1) October 20, (SAMPLING)

6 G.I. Holtzman, Fall 1997 P»15 < X < 25{ = P» -2.5 < Z < 2.5{ = Compute expression (1), P»15 < X < 25{, for samples of size n = 16. P»15 < X < 25{ = P» < Z < 3.33{ = For samples of size n = 4, would expression (1) be smaller, or would it be larger? smaller 5. For samples of size n = 25, would expression (1) be smaller, or would it be larger? larger X-9. Central Limit Theorem (CLT). 1. Rewrite the relationship between the underlying population mean and the mean of the sampling distribution of the mean, which you wrote down earlier. mu X = mu X 2. For this relationship to hold, must the underlying distribution be normal? no 3. If the underlying distribution is not normal, does the sample size have to be large for the relationship to hold? no 4. Does the existence of this relationship depend on the CLT? no 5. Does the location of the sampling distribution of the mean depend on any aspect of the underlying distribution besides its location? no X-10. Consider the relationship between the underlying population standard deviation and the standard error of the sampling distribution of the mean, which you wrote down earlier. 1. Rewrite that formula. sigma X = sigma/sqrtn 2. For this relationship to hold, must the underlying distribution be normal? no 3. If the underlying distribution is not normal, does the sample size have to be large for the relationship to hold? no 4. Does the existence of this relationship depend on the CLT? no 5. Does the dispersion of the sampling distribution of the mean depend on any aspect of the underlying distribution besides its dispersion? no X-11. List all conditions under which the sampling distribution of the mean is exactly normal. if the underlying distribution is normal X-12. List all conditions under which the sampling distribution of the mean is approximately normal. if the sample size is large X-13. The IQ of healthy 18- to 25-year olds is designed to be normally distributed with a mean of 100 and a standard deviation of 15. We model this situation as follows. Let X i be the IQ of the i-th randomly sampled healthy 18- to 25-year old. Assume E(X i ) == mu X = 100 sigma X = 15 X i similar Normal i.e., X i similar N(100, 225). If X denotes the sample mean for samples of size n = 25 then 1. the expected value of the sample mean is E(X) ==mu X =? and the standard error of the mean is sigma X =? 3 October 20, (SAMPLING)

7 Sampling Theory 3. Compute the probability that a randomly selected healthy 18- to 25-year old has an IQ within 5 of average, i.e., between 95 and Compute the probability that the mean of a sample of size 25 is within 5 of the expected value, i.e., between 95 and Compute the standard error of the mean for samples of size Compute the probability that the mean of a sample of size 225 is within 5 of the expected value, i.e., between 95 and As the sample size increases, does the standard error of the mean increase or decrease. decrease 8. As the standard error of the mean decreases, does the probability that the sample mean is near the population mean increase or decrease? increase 9. True or false: The Law of Large Numbers. As the sample size increases, the probability that the sample mean is near the population mean increases. True 10. True or false: The Law of Large Numbers. As the sample size increases, the probability that the sample mean is far from the population mean decreases. True The Law of large numbers is the basis of statistical inference, as you will see in the next exercise. X-14. The IQ of healthy 18- to 25-year olds is designed to be normally distributed with a mean of 100 and a standard deviation of 15. A group of to 25-year olds whose mothers had been heavy drinkers of alcohol during pregnancy were selected to test the hypothesis that children of such mothers would have below-average intelligence. They were found to have an average IQ of Identify the underlying random variable of interest in this study? 2. Indentify the underlying parameter of interest. 3. Compute the expected value of the underlying random variable of interest, assuming that a mother's drinking during pregnancy has no effect Still assuming that the drinking has no effect, compute the probability that one randomly selected individual has an IQ of 97 or less Compute the standard error of the mean for samples of size Still assuming that the drinking has no effect, compute the probability that the mean IQ of 25 randomly selected 18- to 25-year olds is 97 or less Considering that probability, would you infer, based on the result of this study, that 18- to 25-year-old children of mothers who drank heavily during pregnancy are, on the average, less intellegent than normal? I would say no, but it's a matter of opinion. 8. Assuming that the drinking has no effect, compute the standard error of the mean for samples of size Again assuming that the drinking has no effect, compute the probability that the mean IQ of 225 randomly selected 18- to 25-year olds would be 97 or less October 20, (SAMPLING)

8 G.I. Holtzman, Fall Based on that calculation, would you infer, based on a sample of size 225 with an average IQ of 97, that the drinking has a negative effect. Yes 11. For a sample of size 25, what is the greatest mean IQ that would have a lower tail probability of.05 or less? For a sample of size 225, what is the greatest mean IQ that would have a lower tail probability of.05 or less? Which test is more sensitive, the test of size 25, or the test of size 225? Is that not what you would expect? The test with the larger sample size is more sensitive, as one would expect. 14. As sample size increases, does the standard error increase or decrease? decrease 15. As the standard error decreases, does sensitivity increase or decrease? increase 16. As the sample size increases, does sensitivity increase or decrease? increase October 20, (SAMPLING)