Interval Estimation of a Binomial Probability

Interval Estimation of a Binomial ProbabilityIn a random sample of n = 900 registered voters, 475 prefer James for governor.A) What is the point estimate of the population proportion of registered votes preferring James?B) What is the estimate of the standard deviation of the sampling distribution of the sampleproportion (“p-hat”)?C) The “margin of error” for a 95% con?dence interval is 1.96 times this standard deviation.Here the margin of error is?D) The 95% con?dence interval is the point estimate, plus or minus the margin of error. What isthe 95% con?dence interval for the population proportion of individuals preferring JamesNormal Approximation to the BinomialStatistics released by Plum Computer claim that on an average, one out of every 10 PCs is aPlum. Suppose that 400 PCs are randomly sampled.A) The expected number of Plum Computers in the sample is?B) The standard deviation of the number of Plum Computers is?C) What is the probability that the number of Plum Computers will be more than 49 (/i.e.,/ atleast 50)? In using the Normal approximation, use the number 49.5.D) What is the probability that the number of Plum Computers will be more than 50?E) What is the probability that the number of Plum Computers will be exactly 40? Hint:Compute the area over the interval (39.5, 40.5) under the appropriate Normal curve. Normal DistributionsSuppose that the ?ll-volume of soda bottles is Normally distributed with a mean of 10.0 oz. and astandard deviation of 0.25 oz.A) What is the probability that the ?ll-volume is within one standard deviation of its mean?B) What is the probability that the ?ll volume is between 10.0 and 10.125 oz.? Random Sampling for Measured CharacteristicsIn quality control, samples are selected from a production line and various quality characteristicsare measured in order to check that the process is ”in control.” Suppose that a bottling process isintended to ?ll bottles with, on average, 21 ?uid ounces of beverage. Variation around this meanfollows the normal distribution with a standard deviation of 0.5 ?uid ounces. A) If a technician samples 25 bottles (when the process is ”in control”) and measures the amountof beverage in each, what is the standard deviation of the sample mean?B) What is the probability that the sample mean (for the 25 bottles) will exceed 21.2 ?uidounces?C) Find c so that the probability that the sample mean exceeds c when the process is in controlis .05.Exceeding a Speci?cation LimitSuppose that the speci?cation limits for diameters of bolts are 82 mm to 118 mm. Suppose thatthe diameters of individual bolts follow a Normal distribution with a mean of 100 mm and astandard deviation of 6 mm.A) What is the probability that a bolt is within specs?B) Suppose that the manufacturer manages to improve the process so that the standard deviationis only 3 mm. But suppose the mean of the process slips up to 109 mm. What proportion ofdiameters will now exceed the upper speci?cation limit?Computation of the standard deviationFor the sample 4,4,2,9,7,9,7,2,3A) Compute the sum.B) Compute the sum of squares.C) Compute the sum of squared deviations as: D) (a) Compute the variance.(b) Compute the standard deviation.