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The t Distribution Confidence Interval for μ Using the t Distribution

The t Distribution Confidence Interval for μ Using the t Distribution Testing for μ Using the t Distribution. The t Distribution. Conditions Under Which the t Distribution Is Used to Make a Confidence Interval About μ

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The t Distribution Confidence Interval for μ Using the t Distribution

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  1. The t Distribution • Confidence Interval for μUsing the t Distribution • Testing for μUsing the t Distribution

  2. The t Distribution • Conditions Under Which the t Distribution Is Used to Make a Confidence Interval About μ • The t distribution is used to make a confidence interval about μ if • The population from which the sample is drawn is (approximately) normally distributed. • The population standard deviation, σ, unknown.

  3. The t Distribution cont. • The t distribution is a specific type of bell-shaped distribution with a lower height and a wider spread than the standard normal distribution. As the sample size becomes larger, the t distribution approaches the standard normal distribution. The t distribution has only one parameter, called the degrees of freedom (df). The mean of the t distribution is equal to 0 and its standard deviation is .

  4. The t distribution for df = 9 and the standard normal distribution. The standard deviation of the t distribution is The standard deviation of the standard normal distribution is 1.0 μ= 0

  5. Example • Find the value of t for 16 degrees of freedom and .05 area in the right tail of a t distribution curve.

  6. Determining t for 16 df and .05 Area in the Right Tail Area in the right tail df The required value of t for 16 df and .05 area in the right tail

  7. The value of t for 16 df and .05 area in the right tail. .05 df = 16 0 1.746 t This is the required value of t

  8. The value of t for 16 df and .05 area in the left tail. df = 16 .05 -1.746 t 0

  9. Confidence Interval for μUsing the t Distribution • The (1 – α)100% confidence interval for μ is • The value of t is obtained from the t distribution table for n – 1 degrees of freedom and the given confidence level.

  10. Examples …

  11. HYPOTHESIS TEST ABOUT A POPULATION MEAN: SMALL SAMPLES • Test Statistic • The value of the test statistic tfor the sample mean x is computed as • The value of t calculated for x by using this formula is called the observed value of t.

  12. Examples …

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