Synthetic Gem
Statistics Question?
"A new process for the manufacture of synthetic gems produced six stone weighing .43, .52, .46,, .59, .60, and, 56 carats, respectively, in the first run. Find a 90% confidence interval estimate for the average carat weight for this process. "I know the interval be found by using the formula x + / - Z (q / sqrt n). Problem is I'm not the standard ... What should I do? Is there another way they want it from me? Thanks to work.
The traditional definition of the form formula for the standard deviation s = √ [Σ (xx bar) ^ 2 / (n-1)] since x = 0.43, 0.52, 0.46, 0.59, 0.60 and 0.56. X-bar = (0.43 0.52 0.46 0.59 0.60 0.56) / 6 x-bar Σx ≈ 0.5267 = 0.43 0.52 0, 46 +0.59 +0.60 +0.56 = 3.16 Σx ^ 2 = (0.43) ^ 2 + (0.52) ^ 2 + (0.46) ^ 2 + (0.59) ^ 2 + (0.60) ^ 2 + (0.56) ^ 2 = 1.6886 The following form of the following formula to calculate s (the sample standard deviation of the first run of the new process): s = √ ((1 / (n-1) * [Σx ^ 2 - ((Σx) ^ 2 / n)]) s = √ ((1 / (6-1) * [1.6886 - ((3.16) ^ 2 / 6)]) s = 0.069761498 s ≈ 0.0698 Suppose that the data sample distribution is approximately normal divided, We must use the t-distribution (normal curve more diffuse and less focused on the population mean than that of the standard normal curve z-value) due to the sample of n = 6 is less than 30, according to the central limit theorem. Let ∞ be the confidence intervals (the two points ended or tails of the normal curve). The coefficient confidence = 90%, ie (1 - ∞) =% 90% 1 - ∞ ∞ = 0.9 = 0.1 in each end point = ∞ / 2 = 0.1 / 2 = 0.05 t ∞ / 2, (n-1) = T0.05, (6-1) = t0.05, 5 = 2.015 Then, a 90% confidence interval estimate for the average carat weight for this process = [x-bar (+ -) t ∞ / 2, (n -1) * (s / √ n)] = [0.5267 (+ -) t0.05, (6-1) * (0.0698 / √ 6)] = [0.5267 ( + -) 2.015 * (0.0698 / √ 6)] = [0.5267 (+ -) 0.0574] = [0.4693, 0.5841] Therefore, the above interval indicates that we are 95% confident that the true or population mean carat weight the stones for this process is to be between 0.4693 to 0.5841 carats.
