The Central Limit Theorem for Sample Means (Averages) Quiz

PDF page 423: ̄ If you draw random samples of size n, then as n increases, the random variable X , which consists of sample means, tends to be normally distributed and ̄

PDF page 423: ⎛ σ ⎞ x X ∼ N μx, ⎝ 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. 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 − μx z = ,

PDF page 423: ⎛ σ ⎞ x X ∼ N μx, ⎝ 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. 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 − μx z = ,

PDF page 423: ⎛ σ ⎞ x X ∼ N μx, ⎝ 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. 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 − μx z = ,

PDF page 423: ⎛ σ ⎞ x X ∼ N μx, ⎝ 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. 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 − μx z = ,

PDF page 424: ⎝ n⎠ ⎛ 15 ⎞ value = 90 + 2 = 96. ⎝ 25⎠ The value that is two standard deviations above the expected value is 96. 15 σx The standard error of the mean is = = 3. Recall that the standard error of the mean is a description of 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 424: ⎛ σ ⎞ x X ∼ N μx, ⎝ n⎠ ̄ 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 424: ̄ Find P(85 < x < 92). Draw a graph. ̄ P(85 < x < 92) = 0.6997

PDF page 424: b. Find the value that is two standard deviations above the expected value, 90, of the sample mean.

PDF page 424: ⎝ n⎠ ⎛ 15 ⎞ value = 90 + 2 = 96. ⎝ 25⎠ The value that is two standard deviations above the expected value is 96. 15 σx The standard error of the mean is = = 3. Recall that the standard error of the mean is a description of 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 425: ⎝ 50⎠ ̄ Find P(1.8 < x < 2.3). Draw a graph. ̄ P(1.8 < x < 2.3) = 0.9977 normalcdf ⎛ .5 ⎞ 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 426: a. Because the sample mean tends to target the population mean, we have μχ = μ = 34. The sample standard

PDF page 426: d. Let k = the 95 percentile.

PDF page 426: ⎛ 15 ⎞ k = invNorm 0.95,34, = 36.5 ⎝ 100⎠

PDF page 426: b. The central limit theorem states that for large sample sizes (n), the sampling distribution will be

PDF page 426: approximately normal.

PDF page 427: Solution 7.4 σ 1 = 0.13

PDF page 427: a. μ ̄ = μ = 8.2 σ ̄ = =