<< Chapter < Page Chapter >> Page >

Calculators and computers can easily calculate any Student's t-probabilities.

A probability table for the Student's t-distribution can also be used. The table gives t-scores that correspond to the confidence level (column) and degrees of freedom (row). When using a t -table, note that some tables are formatted to show the confidence level in the column headings, while the column headings in some tables may show only corresponding area in one or both tails.

A Student's t table gives t -scores given the degrees of freedom and the right-tailed probability. However such a table is very limited. Calculators and computers can easily calculate any Student's t-probabilities.

    The notation for the Student's t-distribution (using T as the random variable) is:

  • T ~ t df where df = n – 1.
  • For example, if we have a sample of size n = 20 items, then we calculate the degrees of freedom as df = n - 1 = 20 - 1 = 19 and we write the distribution as T ~ t 19 .

If the population standard deviation is not known , the error bound for a population mean is:

  • E B M = ( t α 2 ) ( s n ) ,
  • t σ 2 is the t -score with area to the right equal to α 2 ,
  • use df = n – 1 degrees of freedom, and
  • s = sample standard deviation.

The format for the confidence interval is:
( x ¯ E B M , x ¯ + E B M ) .

Suppose you do a study of acupuncture to determine how effective it is in relieving pain. You measure sensory rates for 15 subjects with the results given. Use the sample data to construct a 95% confidence interval for the mean sensory rate for the population (assumed normal) from which you took the data.

  • 8.6
  • 9.4
  • 7.9
  • 6.8
  • 8.3
  • 7.3
  • 9.2
  • 9.6
  • 8.7
  • 11.4
  • 10.3
  • 5.4
  • 8.1
  • 5.5
  • 6.9

Solution b

Using a calculator, one can obtain a 95% confidence interval of (7.3006, 9.1527)

Note

When calculating the error bound, a probability table for the Student's t-distribution can also be used to find the value of t . The table gives t -scores that correspond to the confidence level (column) and degrees of freedom (row); the t -score is found where the row and column intersect in the table.

Try it

You do a study of hypnotherapy to determine how effective it is in increasing the number of hourse of sleep subjects get each night. You measure hours of sleep for 12 subjects with the following results. Construct a 95% confidence interval for the mean number of hours slept for the population (assumed normal) from which you took the data.

8.2; 9.1; 7.7; 8.6; 6.9; 11.2; 10.1; 9.9; 8.9; 9.2; 7.5; 10.5

(8.1634, 9.8032)

The Human Toxome Project (HTP) is working to understand the scope of industrial pollution in the human body. Industrial chemicals may enter the body through pollution or as ingredients in consumer products. In October 2008, the scientists at HTP tested cord blood samples for 20 newborn infants in the United States. The cord blood of the "In utero/newborn" group was tested for 430 industrial compounds, pollutants, and other chemicals, including chemicals linked to brain and nervous system toxicity, immune system toxicity, and reproductive toxicity, and fertility problems. There are health concerns about the effects of some chemicals on the brain and nervous system. [link] shows how many of the targeted chemicals were found in each infant’s cord blood.

79 145 147 160 116 100 159 151 156 126
137 83 156 94 121 144 123 114 139 99

Use this sample data to construct a 90% confidence interval for the mean number of targeted industrial chemicals to be found in an in infant’s blood.

Solution b

Using a calculator, we find the 90% confidence interval is (117.41, 137.49).

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24' conversation and receive update notifications?

Ask