The Central Limit Theorem for Sample Means (Averages) Quiz

PDF page 394: Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution). Using a subscript that matches the random variable, suppose a. = the mean of X b. = the standard deviation of X If you draw random samples of size n, then as n increases, the random variable , which consists of sample means, tends to be normally distributed and

PDF page 395: normal distribution as n, the sample size, increases.

PDF page 394: The central limit theorem for sample means says that if you keep drawing larger and larger samples (such as rolling one, two, five, and finally, ten dice) and calculating their means, the sample means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by the sample size. The variable n is the number of values that are averaged together, not the number of times the experiment is done. To put it more formally, if you draw random samples of size n, the distribution of the random variable , which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a

PDF page 394: The central limit theorem for sample means says that if you keep drawing larger and larger samples (such as rolling one, two, five, and finally, ten dice) and calculating their means, the sample means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by the sample size. The variable n is the number of values that are averaged together, not the number of times the experiment is done. To put it more formally, if you draw random samples of size n, the distribution of the random variable , which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a

PDF page 394: The central limit theorem for sample means says that if you keep drawing larger and larger samples (such as rolling one, two, five, and finally, ten dice) and calculating their means, the sample means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by the sample size. The variable n is the number of values that are averaged together, not the number of times the experiment is done. To put it more formally, if you draw random samples of size n, the distribution of the random variable , which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a

PDF page 395: μX is the average of both X and . = standard deviation of and is called the standard error of the mean.

PDF page 396: The value that is two standard deviations above the expected value is 96. The standard error of the mean is = = 3. Recall that the standard error of the mean is a description of how far (on average) that the sample mean will be from the population mean in repeated simple random samples of size n.

PDF page 395: Find P(85 < < 92). Draw a graph. P(85 < < 92) = 0.6997 The probability that the sample mean is between 85 and 92 is 0.6997.

PDF page 396: The value that is two standard deviations above the expected value is 96. The standard error of the mean is = = 3. Recall that the standard error of the mean is a description of how far (on average) that the sample mean will be from the population mean in repeated simple random samples of size n.

PDF page 396: b. Find the value that is two standard deviations above the expected value, 90, of the sample mean.

PDF page 396: The value that is two standard deviations above the expected value is 96. The standard error of the mean is = = 3. Recall that the standard error of the mean is a description of how far (on average) that the sample mean will be from the population mean in repeated simple random samples of size n.

PDF page 397: The probability that the mean time is between 1.8 hours and 2.3 hours is 0.9977.

PDF page 398: a. Because the sample mean tends to target the population mean, we have μχ = μ = 34. The sample standard deviation

PDF page 398: c. The probability that the sample mean age is more than 30 is given by P(Χ > 30) = normalcdf(30,E99,34,1.5) = 0.9962.