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STAT 100 Lecture 14 :

Drawing Inferences from Large Samples

Nate Strawn

October 27

Last Time...

(^1) The Sampling Distribution of X for a Normal Population

(^2) The Central Limit Theorem

Statistical Inference

Definition Statistical inference deals with drawing conclusions about population parameters from an analysis of the sample data.

The two most important types of inferences: Estimation of parameters Testing of statistical hypothesis

Statistical Inference

Definition Statistical inference deals with drawing conclusions about population parameters from an analysis of the sample data.

The two most important types of inferences: Estimation of parameters Testing of statistical hypothesis

Example 1.Types of Inference: Point Estimation, Interval

Estimation, and Testing Hypotheses

To study the growth of pine trees at an early stage, a nursery worker records 40 measurements of the heights of one-year-old red pine seedlings. This set of measurements is Heights of One-Year-Old Red Pine Seedlings Measured in Centimeters

2.6 1.9 1.8 1.6 1.4 2.2 1.2 1. 1.6 1.5 1.4 1.6 2.3 1.5 1.1 1. 2.0 1.5 1.7 1.5 1.6 2.1 2.8 1. 1.2 1.2 1.8 1.7 0.8 1.5 2.0 2. 1.5 1.6 2.2 2.1 3.1 1.7 1.7 1.

(Courtesy of Professor Alan Ek.)

Example 1.Types of Inference: Point Estimation, Interval

Estimation, and Testing Hypotheses

More specifically, depending on the purpose of the study, we may wish to do one, two, or all three of the following: Estimate a single value for the unknown μ (point estimation). Determine an interval of plausible values for μ (interval estimation). Decide whether or not the mean height μ is 1.9 centimeters, which was previously found to be the mean height of a different stock of pine seedlings (testing statistical hypotheses).

Example 1.Types of Inference: Point Estimation, Interval

Estimation, and Testing Hypotheses

More specifically, depending on the purpose of the study, we may wish to do one, two, or all three of the following: Estimate a single value for the unknown μ (point estimation). Determine an interval of plausible values for μ (interval estimation). Decide whether or not the mean height μ is 1.9 centimeters, which was previously found to be the mean height of a different stock of pine seedlings (testing statistical hypotheses).

Example 1.Types of Inference: Point Estimation, Interval

Estimation, and Testing Hypotheses

More specifically, depending on the purpose of the study, we may wish to do one, two, or all three of the following: Estimate a single value for the unknown μ (point estimation). Determine an interval of plausible values for μ (interval estimation). Decide whether or not the mean height μ is 1.9 centimeters, which was previously found to be the mean height of a different stock of pine seedlings (testing statistical hypotheses).

Example 2. Inferences about an Unknown Proportion

Example (^1) How close can we expect a sample percentage to be to the population percentage? For instance, if 56.5% of 2705 sampled voters supported Schwarzenegger, how close to 56.5% is the percentage of entire population of 7 million voters who voted for him? (^2) The sample proportion ˆp=56.5 % sheds some light on p, but it is subject to some error since it draws only on a part of the population. We would like to evaluate its margin of error and provide an interval of plausible values of p.

Point Estimation of a Population Mean

Definition A statistic intended for estimating a parameter is called a point estimator, or simply an estimator. The standard deviation of an estimator is called its standard error: S.E.

A population mean μ is estimated by sample mean

X =

X 1 + X 2 + · · · + Xn n

The properties of the sample mean X.

When we are approximating μ by X , the 95.4% error margin is 2 σ/

n

The 100(1 − α)% error margin.

Definition zα/ 2 = Upper α/ 2 point of standard normal distribution. That is, the area to the right of zα/ 2 is α/ 2 , and the area between −zα/ 2 and zα/ 2 is 1 − α.

Point Estimation of the Mean.

Definition Parameter: Population mean μ. Data: X 1 , X 2 ,... , Xn (a random sample of size n) Estimator: X (sample mean)

S.E .(X ) = σ/

n

Estimated S.E .(X ) = S/

n For large n, the 100(1 − α)% error margin is zα/ 2 σ/

n. (If σ is unknown, use S in place of σ.)

Point Estimation of the Mean Height of Seedings.

Example From the data of Height of Seedings, consisting of 40 measurements of the heights of one-year-old red pine seedlings, give a point estimate of the population mean height and state a 95 % error margin. Solution x =

xi /40 = 1. 715

s =

xi − x 40 − 1

To calculate the 95% error margin, we set 1 − α =. 95 so that α/ 2 = .025 and zα/ 2 = 1.96. Therefore, the 95% error margin is

1. 96 s/

n = 1. 96 ×. 475 /

40 =. 15 cm