A textile fiber manufacturer is investigating a new drapery yarn, which has a standard deviation of 0.3 kgThe company wishes to test the hypothesis H0: μ 14 against H1: μ lt; 14, using a random sample of 16 specimens.(a) What is the P-value if the sample average is X 13.7 kg? Round your answer to 2 decimal places.(b) Find β for the case where the true mean elongation force is 13.5 kg and we assume that α 0.05Round your intermediate answer to 2 decimal placesRound your final answer to 2 decimal places.(c) What is the power of the test from part (b)? Round your answer to 2 decimal places.
Aluminum's density is about 2.7 grams per cubic centimeterSo a sphere with a mass of 84grams has a volume of about 31.111 cubic centimetersYou get this by dividing 84 by 2.7Now that you know the volume, you can calculate the radius of the spherefor a sphere, volume 4/3 pi radius to the third powerSo 31.111 4/3 (3.14.) r^3Divide both sides by 4/3 then divide both sides by pi and you get something like r^3 is 7.427Then find the cube root of this to find your radiusI don't have a calculator, so someone should check the numbers.
Volume mass density So you can find the volumeThen use V 4/3 pi r^3 as the formula for volume given radiusSet the volume from above, and solve for radiusWatch the final units of your answer, you may have to convert cm to inches or something like that.
Density Of Aluminum In Inches
1) You should know the central limit theorem: Sample averages of size N, 16 in your case, will always have a mean equal to the mean, μ14 in your case, of the original population and a standard deviation equal to the standard deviation of the original population, σ0.3 in your case, divided by √NAlso, for large enough samples, the sample averages will be normally distributed 2) You need to know how to use your normal distribution table???Calculate the Z-score (x - mean) / (standard_deviation) (x-μ)/σThe normal distribution table converts a Z-score to the probability, the P-value, that the sample mean is this small or smallerIf the probability is small, less than or equal to α, you should reject the null hypothesis (H0) and support the alternate hypothesis (H1)- For part (b), β is just the probability of getting X, the sample mean, so high that the P-value for the null hypothesis (H0, μ14) is greater than α (5%) - generating a Type II error, not rejecting the Null Hypothesis when you shouldUse your normal distribution table again, working back from a P-valueα to get a z-score, to get Xμ+zσThen calculate the z-score for that X using the TRUE population mean, 13.5 [Z(X-13.5)/0.3]From that second z-score you can calculate the probability (from that table again) that the actual sample mean is LESS than this value, a new P-valueBut you need the probability that the sample mean is MORE than that X: β1-(P-value) - Part (c), the power of the test, is just the probability that the the test does NOT make a type II error: 1-βThe second P-value from part (b).
