The Central Limit Theorem for Sample Means (Averages) Quiz

PDF page 408: ̄ If you draw random samples of size n, then as n increases, the random variable X which consists of sample means, tends

PDF page 409: ⎛ σx⎞ . X ~ N μ x , ⎝ n⎠ 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. Standard deviation is the square root of variance, so the standard deviation of the sampling distribution is the standard deviation of the original distribution divided by the square root of n. 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 X , which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a normal distribution as n, the sample size, increases. ̄ ̄ The random variable X has a different z-score associated with it from that of the random variable X. The mean x is the ̄ value of X in one sample. ̄ x − μ x z =

PDF page 409: ⎛ σx⎞ . X ~ N μ x , ⎝ n⎠ 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. Standard deviation is the square root of variance, so the standard deviation of the sampling distribution is the standard deviation of the original distribution divided by the square root of n. 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 X , which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a normal distribution as n, the sample size, increases. ̄ ̄ The random variable X has a different z-score associated with it from that of the random variable X. The mean x is the ̄ value of X in one sample. ̄ x − μ x z =

PDF page 409: ⎛ σx⎞ . X ~ N μ x , ⎝ n⎠ 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. Standard deviation is the square root of variance, so the standard deviation of the sampling distribution is the standard deviation of the original distribution divided by the square root of n. 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 X , which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a normal distribution as n, the sample size, increases. ̄ ̄ The random variable X has a different z-score associated with it from that of the random variable X. The mean x is the ̄ value of X in one sample. ̄ x − μ x z =

PDF page 409: ⎛ σx⎞ . X ~ N μ x , ⎝ n⎠ 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. Standard deviation is the square root of variance, so the standard deviation of the sampling distribution is the standard deviation of the original distribution divided by the square root of n. 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 X , which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a normal distribution as n, the sample size, increases. ̄ ̄ The random variable X has a different z-score associated with it from that of the random variable X. The mean x is the ̄ value of X in one sample. ̄ x − μ x z =

PDF page 409: ⎛ σx⎞ . X ~ N μ x , ⎝ n⎠ 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. Standard deviation is the square root of variance, so the standard deviation of the sampling distribution is the standard deviation of the original distribution divided by the square root of n. 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 X , which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a normal distribution as n, the sample size, increases. ̄ ̄ The random variable X has a different z-score associated with it from that of the random variable X. The mean x is the ̄ value of X in one sample. ̄ x − μ x z =

PDF page 409: = standard deviation of X and is called the standard error of the mean. n

PDF page 410: ̄ ⎛ 15 ⎞ X ~ N 90, . ⎝ 25⎠ ̄ Find P(85 < x < 92). Draw a graph. ̄ P(85 < x < 92) = 0.6997 The probability that the sample mean is between 85 and 92 is 0.6997.

PDF page 410: normalcdf(lower value, upper value, mean, standard error of the mean) σ The parameter list is abbreviated (lower value, upper value, μ, ) n 15 normalcdf(85,92,90, ) = 0.6997 25

PDF page 410: ⎝ n⎠ ⎛ 15 ⎞ value = 90 + 2 = 96 ⎝ 25⎠ The value that is two standard deviations above the expected value is 96. 15 σx = = 3. Recall that the standard error of the mean is a description of The standard error of the mean is n 25 how far (on average) that the sample mean will be from the population mean in repeated simple random samples of size n.

PDF page 410: value = μx + (#ofTSDEVs)

PDF page 410: ⎝ n⎠ ⎛ 15 ⎞ value = 90 + 2 = 96 ⎝ 25⎠ The value that is two standard deviations above the expected value is 96. 15 σx = = 3. Recall that the standard error of the mean is a description of The standard error of the mean is n 25 how far (on average) that the sample mean will be from the population mean in repeated simple random samples of size n.

PDF page 409: • mean is the mean of the original distribution

PDF page 411: μX = 2, σX = 0.5, n = 50, and X ~ N 2,

PDF page 411: ⎝ 50⎠ ̄ Find P(1.8 < x < 2.3). Draw a graph. ̄ P(1.8 < x < 2.3) = 0.9977 ⎛ .5 ⎞ normalcdf 1.8,2.3,2, = 0.9977 ⎝ 50⎠ The probability that the mean time is between 1.8 hours and 2.3 hours is 0.9977.

PDF page 411: standard deviation⎞ k = invNorm area to the left of k, mean,

PDF page 411: To find percentiles for means on the calculator, follow these steps.