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Assumptions for Inference About a Mean
Our data are a simple random sample (SRS) of size n from the population.
Observations from the population have a normal distribution with mean μand standard deviation σ. Both μand σ are unknown parameters. In the previous chapter we made the
unrealistic assumption that we knew the value of σ, when in practice σ is unknown.
Standard Error
Because we don’t know σ, we estimate it by the sample standard deviation s.
When the standard deviation of a statistic is estimated from the data, the result is called the standard error of the statistic. The standard error of the sample mean is :
xs
n
The One-Sample t Statistic and the t Distributions
Draw an SRS of size n from a population that has the normal distribution with mean μ and standard deviation σ.
The one-sample t statistic
has the t distribution with n – 1 degrees of freedom.
t =x−μ
sn
Facts About t Distributions
The density curves of the t distributions are similar in shape to the standard normal curve. They are symmetric about zero and are bell-shaped.
The spread of the t distributions is a bit greater than that of the standard normal distribution. This comes from using s instead of σ.
As the degrees of freedom increases, the density curve approaches the standard normal curve.
t chart Examples
What critical values from Table C satisfies each of the following conditions?
A. The t distribution with 8 degrees of freedom has probability 0.025 to the right of t*
B. The t distribution with 17 degrees of freedom has probability 0.20 to the left of t*
C. The one-sampled t statistics from a sample of 25 observations has probability 0.01 to the right of t*.
D. The one-sampled t statistics from an SRS of 30 observations has probability 0.95 to the left of t*.
Example
The one-sample t statistic for testing H0: μ= 0 Ha: μ> 0
From a sample of 10 observations has the value t = 3.12 A. What are the degrees of freedom for this statistic? B. Give the two critical values of t* from the Table C
from bracket t. C. Between what two values does the P-value of this
test fall? D. Is the value t = 3.12 significant at the 5% level?
Is it significant at the 1% level?