St at is t ic s 270 - Lec t ure 22

• Last Day…completed 5.1

• Today Parts of Section 5.3 and 5.4

Ex am ple

• 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 I nference 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 Est im at ion

• 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:

• I f the sampling procedure is repeated many times, what proportion of sample means lie in the interval:

• In general, 100(1- )% of sample means fall in the interval

• Therefore, before sampling the probability of getting a sample mean in this interval is

nz

nz 2/2/ ,

• Could write this as:

• Or, re-writing…we get:

)1(2/2/ nzX

nzP

)1(2/2/ nzX

nzXP

• The interval below is called a confidence interval for

• Key features:• Population distribution is assumed to be normal• Population standard deviation, , is known

nzX

nzX 2/2/ ,

Ex am ple

• 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

In t erpret at ion

• 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 Sam ple Conf idenc e In t erva l for

• Situation:

• Have a random sample of size n (large)

• Suppose value of the standard deviation is known

• Value of population mean is unknown

• I f 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 confidence interval for the mean is:

)%1(100

Ex am ple

• 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 t he Lengt h of a Conf idenc e In t erva l

• Can shorten the length of a confidence interval by:

• Using a difference confidence level• Increasing the sample size

• Reducing population standard deviation

Sam ple Size for a Desi red Widt h

• 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

Sam ple Size for a Desi red Widt h

• Width (need to specify confidence level)

• Sample size for the desired width

Ex am ple

• Limnologists wishes to estimate the mean phosphate content per unit volume of a lake water

• I t is known from previous studies that the standard deviation isfairly 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?

Ex am ple

• 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?

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