The t-DistributionEstimating a Population Mean

t-Distribution

When sample sizes are sometimes small, and often we do not know the standard deviation of the population, statisticians rely on the distribution of the t statistic (also known as the t score), whose values are given by:

X ± t* √n

_ _s

Degrees of freedom

There are actually many different t distributions. The particular form of the t distribution is

determined by its degrees of freedom. The degrees of freedom refers to the number of independent observations in a set of data.

df = n-1

When to Use the t-Distribution

The population distribution is normal.The sampling distribution is symmetric, unimodal, without outliers, and the sample size is 15 or less.The sampling distribution is moderately skewed, unimodal, without outliers, and the sample size is between 16 and 40.The sample size is greater than 40, without outliers.

The t distribution should not be used with small samples from populations that are not

approximately normal.

Auto polutionConstructing a one-sample t-interval for ℳ

Environmentalists, government officials, and vehicle

manufactureres are all interested in studying the auto exhaust emissions produced by motor

vehicles. The major pollutants in auto exhaust from gasoline engines are hydorcarbon,

monoxide, and nitrogen oxides.

1.28 1.17 1.16 1.08 0.60 1.321.24 0.71 0.49 1.38 1.20 0.780.95 2.20 1.78 1.83 1.26 1.731.31 1.80 1.15 0.97 1.12 0.721.31 1.45 1.22 1.32 1.47 1.440.51 1.49 1.33 0.86 0.57 1.792.27 1.87 2.94 1.16 1.45 1.511.47 1.06 2.01

Amount of nitrogen oxides (NOX) emitted by light-duty engines

(grams/mile)

Construct a 95% confidence interval for the mean amount of NOX emitted by light-duty engines of this

type.x =

1.329

n = 45

df = 44

s =

0.484

t* =

2.021

_

t confidence interval: (1.185, 1.473)

We are 95% confident that the true mean level of nitrogen

oxides emitted by this type of light-duty engine is between 1.185 and 1.473 grams/mole

X ± t* √n

_ _s

The Student t-distribution The t-distribution is a family of distributions indexed by a

“degrees of freedom” parameter. (Different distribution for different sample sizes)

The degrees of freedom is the number of sample values that can vary after certain restrictions have been imposed on all data values.

It has a bell shape, but has greater variability than N(0,1). (has a wider spread.)

As the sample size n gets larger, the Student t- distribution gets closer to the normal distribution. When the sample size is small, the degrees of freedom is small and there is more variability (i.e., wider spread).

When the sample size is large, the degrees of freedom is large and there is less variability, and it’s closer to N(0,1).

The mean is 0. The standard deviation is greater than 1.

Density curves for t distribution

Standard Error of point estimateStandard Error of a point estimate is a common term for

standard deviation of the point estimateStandard error of y: (based on known σ)

Clarification of terminologyAn estimator is a rule for computing a quantity

from a sample that is to be used to estimate a model parameter.

An estimate is the value that the rule gives when the data are taken.

The distribution of the estimator is called its sampling distribution.

The standard deviation of the sampling distribution of an estimator is called the standard error of the estimator.

Properties of the t-distributions:

1.The t-curve corresponding to any fixed number of degrees of freedom (df) is bell shaped, symmetric and centered at 0.

2.Each t-curve is more spread out than the z-curve (standard normal curve).

3.As the df increase, the spread of the corresponding t-curve decreases.

4.As the number of df increases, the t-curves get closer and closer to the z-curve.

5.When estimating a single mean, df = n – 1.

Estimating a population mean