sampling distributions & interval estimation · the sampling distribution of the sample mean...

Sampling Distributions & Interval Estimation

W. Rofianto

Parameters

▪ The director of personnel for Electronics Associates, Inc. (EAI), has beenassigned the task of developing a profile of the company’s 2500 managers.

▪ The characteristics to be identified include the mean annual salary for themanagers and the proportion of managers having completed the company’smanagement training program.

▪ Numerical characteristics of a population are called parameters.

▪ Suppose that the necessary information on all the EAI managers was notreadily available in the company’s database.

▪ The question we now consider is how the firm’s director of personnel canobtain estimates of the population parameters by using a sample ofmanagers rather than all 2500 managers in the population.

Sampling from a Finite Population

▪ Statisticians recommend selecting a probability sample when sampling froma finite population because a probability sample allows them to make validstatistical inferences about the population.

▪ First step is construct a frame by assigning each manager a number. Forexample, we can assign the managers the numbers 1 to 2500 .

▪ 6327 1599 8671 7445 1102 1514 1807 so forth

Point Estimation

Point Estimation

Sampling Distributions

▪ Suppose we repeat the process of selecting a simple random sample of 30EAI managers over and over again, each time computing the values of andTable 7.4 contains a portion of the results obtained for 500 simple randomsamples.

Sampling distribution of

Expected Value & Standard Deviation of

(standard error)

Form of the Sampling Distribution of

CENTRAL LIMIT THEOREM

In selecting random samplesof size n from a population,the sampling distribution ofthe sample mean can beapproximated by a normaldistribution as the samplesize becomes large.

Sampling distribution of

Probability Of a Sample Mean

▪ .

z

Sampling Distribution of

Interval Estimation

▪ Because a point estimator cannot be expected to provide the exact value ofthe population parameter, an interval estimate is often computed by addingand subtracting a value, called the margin of error, to the point estimate.The general form of an interval estimate is as follows:

▪ The purpose of an interval estimate is to provide information about howclose the point estimate, provided by the sample, is to the value of thepopulation parameter.

Interval Estimate Of A Population Mean: σ Known

▪ Each week Lloyd’s Department Store selects a simple random sample of 100customers. Lloyd’s assumes a known value of σ=$20, and obtained a samplemean of =$82.

We say that this intervalhas been established atthe 95% confidence level.The value .95 is referred toas the confidencecoefficient, and theinterval 78.08 to 85.92 iscalled the 95% confidenceinterval.

Interval Estimate Of A Population Mean: σ Known

▪ Each week Lloyd’s Department Store selects a simple random sample of 100customers. Lloyd’s assumes a known value of σ=$20, and obtained a samplemean of =$82.

We say that this interval has beenestablished at the 95% confidence level.The value .95 is referred to as theconfidence coefficient, and the interval78.08 to 85.92 is called the 95%confidence interval.

Interval Estimate Of A Population Mean: σ Unknown

▪ When s is used toestimate σ (σ unknowncase), the margin of errorand the interval estimatefor the population meanare based on a probabilitydistribution known as thet distribution.

Z vs t Distribution

▪ .

Determining the Sample Size

▪ The project director specified a desired margin of error of E=2, and the 95%level of confidence indicates z.025=1.96. An analyst reviewed the sample datafrom the previous study and found that the sample standard deviation forthe daily rental cost was $9.65.

▪ The sample size for the new study needs to be at least:

Population Proportion

▪ .

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