A Single Population Mean Using the Normal Distribution Quiz

PDF page 469: To construct a confidence interval for a single unknown population mean, μ, where the population standard deviation is ̄ known, we need x as an estimate for μ, and we need the margin of error. Here, the margin of error is called the error bound for a population mean (EBM) is called the margin of error for a population mean (EBM). The sample mean, ̄ x , is the point estimate of the unknown population mean, μ. The confidence interval (CI) estimate will have the form: ̄ ̄ (point estimate – error bound, point estimate + error bound) or, in symbols, ( x – EBM, x +EBM ). The margin of error (EBM) depends on the confidence level (CL). The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. However, it is more accurate to state that the confidence level is the percentage of confidence intervals that contain the true population parameter when repeated samples are taken. Most often, the person constructing the confidence interval will choose a confidence level of 90 percent or higher, because that person wants to be reasonably certain of his or her conclusions. Another probability, which is called alpha (α) is related to the confidence level, CL. Alpha is the probability that the confidence interval does not contain the unknown population parameter. Mathematically, alpha can be computed as

PDF page 471: • When the population standard deviation σ is known, we use a normal distribution to calculate the error bound.

PDF page 470: Suppose we have collected data from a sample. We know the sample mean, but we do not know the mean for the entire population. The sample mean is seven, and the error bound for the mean is 2.5. ̄ x and EBM = 2.5. The confidence interval is (7 – 2.5, 7 + 2.5), and calculating the values gives (4.5, 9.5). If the confidence level is 95 percent, then we say, "We estimate with 95 percent confidence that the true value of the population mean is between 4.5 and 9.5."

PDF page 470: A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means ̄ follow an approximately normal distribution. Suppose that our sample has a mean of x = 10, and we have constructed the 90 percent confidence interval (5, 15) where EBM = 5. To get a 90 percent confidence interval, we must include the central 90 percent of the probability of the normal distribution. If we include the central 90 percent, we leave out a total of α = 10 percent in both tails, or 5 percent in each tail, of the normal distribution.

PDF page 470: The critical value 1.645 is the z-score in a standard normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail. To capture the central 90 percent, we must go out 1.645 standard deviations on either side of the calculated sample mean. The critical value will change depending on the confidence level of the interval. It is important that the standard deviation used be appropriate for the parameter we are estimating, so in this section, we σ σ need to use the standard deviation that applies to sample means, which is . The fraction is commonly called the n n standard error of the mean in order to distinguish clearly the standard deviation for a mean from the population standard deviation, σ. In summary, as a result of the central limit theorem, the following statements apply: ̄ ̄

PDF page 470: The critical value 1.645 is the z-score in a standard normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail. To capture the central 90 percent, we must go out 1.645 standard deviations on either side of the calculated sample mean. The critical value will change depending on the confidence level of the interval. It is important that the standard deviation used be appropriate for the parameter we are estimating, so in this section, we σ σ need to use the standard deviation that applies to sample means, which is . The fraction is commonly called the n n standard error of the mean in order to distinguish clearly the standard deviation for a mean from the population standard deviation, σ. In summary, as a result of the central limit theorem, the following statements apply: ̄ ̄

PDF page 469: To construct a confidence interval for a single unknown population mean, μ, where the population standard deviation is ̄ known, we need x as an estimate for μ, and we need the margin of error. Here, the margin of error is called the error bound for a population mean (EBM) is called the margin of error for a population mean (EBM). The sample mean, ̄ x , is the point estimate of the unknown population mean, μ. The confidence interval (CI) estimate will have the form: ̄ ̄ (point estimate – error bound, point estimate + error bound) or, in symbols, ( x – EBM, x +EBM ). The margin of error (EBM) depends on the confidence level (CL). The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. However, it is more accurate to state that the confidence level is the percentage of confidence intervals that contain the true population parameter when repeated samples are taken. Most often, the person constructing the confidence interval will choose a confidence level of 90 percent or higher, because that person wants to be reasonably certain of his or her conclusions. Another probability, which is called alpha (α) is related to the confidence level, CL. Alpha is the probability that the confidence interval does not contain the unknown population parameter. Mathematically, alpha can be computed as

PDF page 471: Calculating the Margin of Error EBM The error bound formula for an unknown population mean, μ, when the population standard deviation, σ, is known is

PDF page 476: CL = 0.95, so α = 1 – CL = 1 – 0.95 = 0.05.

PDF page 471: 2 The area to the right of z0.025 is 0.025 and the area to the left of z0.025 is 1 – 0.025 = 0.975. = 1.96 , using a calculator, computer, or standard normal probability table. z α = z 0.025 2 Normal table (see appendices) shows that the probability for 0 to 1.96 is 0.47500, and so the probability to the right tail of the critical value 1.96 is 0.5 – 0.475 = 0.025

PDF page 477: = 1.96, z α = z 0.025 2 when using invnorm(0.975,0,1) on the TI-83, 83+, or 84+ calculators. (This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the standard normal distribution.) ⎛ 3 ⎞ EBM = (1.96) = 0.98 ⎝ 36⎠ ̄

PDF page 473: Solution 8.2 Solution B

PDF page 473: Press STAT and arrow over to TESTS. Arrow down to 7:ZInterval. Press ENTER. Arrow to Stats and press ENTER. ̄ Arrow down and enter 3 for σ, 68 for x , 36 for n, and .90 for C-level. Arrow down to Calculate and press ENTER. The confidence interval is (to three decimal places)(67.178, 68.822).

PDF page 477: = 1.96, z α = z 0.025 2 when using invnorm(0.975,0,1) on the TI-83, 83+, or 84+ calculators. (This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the standard normal distribution.) ⎛ 3 ⎞ EBM = (1.96) = 0.98 ⎝ 36⎠ ̄

PDF page 477: The 90 percent confidence interval is (67.18, 68.82). The 95 percent confidence interval is (67.02, 68.98). The 95 percent confidence interval is wider. If you look at the graphs, because the area 0.95 is larger than the area 0.90, it makes sense that the 95 percent confidence interval is wider. For more certainty that the confidence interval actually does contain the true value of the population mean for all statistics exam scores, the confidence interval necessarily needs to be wider.