Transcript Lecture 22
Statistics 270 - Lecture 22 • Last Day…completed 5.1 • Today Parts of Section 5.3 and 5.4 Example • Government regulations indicate that the total weight of cargo in a certain kind of airplane cannot exceed 330 kg. On a particular day a plane is loaded with 81 boxes of a particular item only. Historically, the weight distribution for the individual boxes of this variety has a mean 3.2 kg and standard deviation 1.0 kg. • What is the distribution of the sample mean weight for the boxes? • What is the probability that the observed sample mean is larger than 3.33 kg? • Statistical Inference deals with drawing conclusions about population parameters from sample data • Estimation of parameters: • Estimate a single value for the parameter (point estimate) • Estimate a plausible range of values for the parameter (confidence intervals) • Testing hypothesis: • Procedure for testing whether or not the data support a theory or hypothesis Point Estimation • Objective: to estimate a population parameter based on the sample data • Point estimator is a statistic which estimates the population parameter • Suppose have a random sample of size n from a normal population • What is the distribution of the sample mean? • If the sampling procedure is repeated many times, what proportion of sample means lie in the interval: • If the sampling procedure is repeated many times, what proportion of sample means lie in the interval: • In general, 100(1-a)% of sample means fall in the interval z , z a /2 a /2 n n • Therefore, before sampling the probability of getting a sample mean in this interval is • Could write this as: P za / 2 X za / 2 (1 a ) n n • Or, re-writing…we get: P X za / 2 X za / 2 (1 a ) n n • The interval below is called a confidence interval for X z , X z a /2 a /2 n n • Key features: • Population distribution is assumed to be normal • Population standard deviation, , is known Example • To assess the accuracy of a laboratory scale, a standard weight known to be 10 grams is weighed 5 times • The reading are normally distributed with unknown mean and a standard deviation of 0.0002 grams • Mean result is 10.0023 grams • Find a 90% confidence interval for the mean Interpretation • What exactly is the confidence interval telling us? • Consider the interval in the previous example. What is the probability that the population mean is in that particular interval? • Consider the interval in the previous example. What is the probability that the sample mean is in that particular interval? Large Sample Confidence Interval for • Situation: • Have a random sample of size n (large) • Suppose value of the standard deviation is known • Value of population mean is unknown • If n is large, distribution of sample mean is • Can use this result to get an approximate confidence interval for the population mean • When n is large, an approximate for the mean is: 100(1 a )% confidence interval Example • Amount of fat was measured for a random sample of 35 hamburgers of a particular restaurant chain • It is known from previous studies that the standard deviation of the fat content is 3.8 grams • Sample mean was found to be 30.2 • Find a 95% confidence interval for the mean fat content of hamburgers for this chain Changing the Length of a Confidence Interval • Can shorten the length of a confidence interval by: • Using a difference confidence level • Increasing the sample size • Reducing population standard deviation Sample Size for a Desired Width • Frequent question is “how large a sample should I take?” • Well, it depends • One to answer this is to construct a confidence interval for a desired width Sample Size for a Desired Width • Width (need to specify confidence level) • Sample size for the desired width Example • Limnologists wishes to estimate the mean phosphate content per unit volume of a lake water • It is known from previous studies that the standard deviation is fairly stable at around 4 ppm and that the observations are normally distributed • How many samples must be sampled to be 95% confidence of being within .8 ppm of the true value? Example • A plant scientist wishes to know the average nitrogen uptake of a vegetable crop • A pilot study showed that the standard deviation of the update is about 120 ppm • She wishes to be 90% confident of knowing the true mean within 20 ppm • What is the required sample size?