T Distribution



t Distribution

When [pic] is the mean of a random sample of size n from a normal distribution with mean [pic], the random variable

[pic]

has a probability distribution called a t distribution with n – 1 degrees of freedom (df).

Properties of t Distributions

Let tv denote the density function curve for v df.

1. Each tv curve is bell-shaped and centered at 0.

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

3. As [pic], the sequence of tv curves approaches the standard normal curve (so the z curve is often called the t curve with [pic])

Historical Perspective

The distribution of the t statistic in repeated sampling was discovered by W. S. Gosset, a chemist in the Guinness brewery in Ireland, who published his discovery in 1908 under the pen name of Student.

Confidence Interval for μ

Using a Small Sample ( n[pic])

[pic]

Assumptions:

1. A simple random sample is selected from the population

2. The population distribution is approximately normal.

Hypothesis Test for μ

Using a Small Sample (n[pic]30)

Ho: μ = μo

Ha: 1. μ > μo

2. μ < μo

3. μ [pic] μo

Test Statistic: [pic]

Rejection Region: For a probability α of a Type-I error, we can reject Ho if

1. t [pic] tα

2. t [pic] - tα

3. t [pic] - tα/2 or t [pic] tα/2

Assumptions:

1. A simple random sample is selected from the population

2. The population distribution is approximately normal.

Example A major new car manufacturer wants to test a new engine to determine whether it meets new air-pollution standards. The mean emission μ of all engines of this type must be less than 20 parts per million of carbon. Ten engines are manufactured for testing purposes, and the emission level of each is determined. The data ( in parts per million) are listed below:

15.6 16.2 22.5 20.5 16.4 19.4 16.6 17.9 12.7 13.9

Do the data provide sufficient evidence to allow the manufacturer to conclude that the type of engine meets the pollution standard? Use α=.01.

Assessing Reasonableness of Normality Assumption

Using Minitab

Minitab Commands for Normal Probability Plot: > stat > basic statistics > normality test

[pic] [pic]

Conducting Hypothesis Test Using Minitab

Minitab Commands: > stat > basic statistics > 1-sample t > test mean 102 > options > alternative less than

Minitab Output:

Test of μ = 20 vs < 20

Variable N Mean StDev SE Mean 95% Upper Bound T P

emission level 10 17.170 2.981 0.943 18.898 -3.00 0.007

Constructing Confidence Interval Using Minitab

Example Patients with normal hearing in one ear and unaided sensorineural hearing loss in the other are characterized as suffering from unilateral hearing loss. In a study reported in the American Journal of Audiology (Mar. 1995), eight patients with unilateral hearing loss were fitted with a special wireless hearing aid in the “bad” ear. The absolute sound pressure level (SPL) was then measured near the eardrum of the ear when noise was produced at a frequency of 500 hertz. The SPLs of the eight patients, recorded in decibels, are listed below.

73.0 80.1 82.8 76.8 73.5 74.3 76.0 68.1

Construct a 90% confidence interval for the population mean SPL of unilateral hearing loss patients when noise is produced at a frequency of 500 hertz.

Assessing Reasonableness of Normality Assumption

Using Minitab

[pic]

Constructing Confidence Interval

Using Minitab

Minitab Commands: > stat > basic statistics > 1-sample t > options > confidence level > 90 > alternative not equal

Minitab Output:

One-Sample T: spl

Variable N Mean StDev SE Mean 90% CI

spl 8 75.58 4.52 1.60 (72.55, 78.60)

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