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Chapter 11. Section 11.1 – Inference for the Mean of a Population. Inference for the Mean of a Population.

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chapter 11

Chapter 11

Section 11.1 – Inference for the Mean of a Population

inference for the mean of a population
Inference for the Mean of a Population
  • Confidence intervals and tests of significance for the mean μ of a normal population are based on the sample mean . The sampling distribution of has μ as its mean. That is an unbiased estimator of the unknown μ.
  • In the previous chapter we make the unrealistic assumption that we knew the value of σ. In practice, σ is unknown.
conditions for inference about a mean
Conditions for Inference About a Mean
  • Our data are a simple random sample (SRS) of size n from the population of interest. This condition is very important.
  • Observations from the population have a normal distribution with mean μ and standard deviation σ. In practice, it is enough that the distribution be symmetric and single-peaked unless the sample is very small.
  • Both μ and σ are unknown parameters.
standard error
Standard Error
  • 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 .
the t distributions
The t distributions
  • When we know the value of σ, we base confidence intervals and tests for μ on one-sample z statistics
  • When we do not know σ, we substitute the standard error of for its standard deviation .
  • The statistic that results does not have a normal distribution. It has a distribution that is new to us, called a t distribution.
t distributions continued
t distributions (continued…)
  • The density curves of the t distributions are similar in shape to the standard normal curve. They are symmetric about zero, single-peaked, and bell shaped.
  • The spread of the t distribution is a bit greater than that of the standard normal distribution. The t have more probability in the tails and less in the center than does the standard normal.
  • As the degrees of freedom k increase, the t(k) density curve approached the N(0,1) curve ever more closely.
the one sample t procedures
The One-Sample t Procedures
  • Draw an SRS of size n from a population having unknown mean μ. A level C confidence interval for μ is
  • Where is the upper (1 - C)/2 critical value for the t(n– 1) distribution. This interval is exact when the population distribution is normal and is approximately correct for large n in other cases.
  • The test the hypothesis H0 : μ = μ0 based on an SRS of size n, computed the one-sample t statistic
degrees of freedom
Degrees of Freedom
  • There is a different t distribution for each sample size. We specify a particular t distribution by giving its degree of freedom.
  • The degree of freedom for the one-sided t statistic come from the sample standard deviation s in the denominator of t.
  • We will write the t distribution with k degrees of freedom as t(k) for short.
example 11 1 using the t table
Example 11.1 - Using the “t Table”
  • What critical value t* from Table C (back cover of text book, often referred to as the “t table”) would you use for a t distribution with 18 degrees of freedom having probability 0.90 to the left of t?
  • Now suppose you want to construct a 95% confidence interval for the mean of a population based on an SRS of size n = 12. What critical value should you use?
the one sample t statistic and the t distribution
The One-sample t Statistic and the t Distribution
  • 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.
the one sample t procedure continued
The One-sample t Procedure (continued…)
  • In terms of a variable T having then

t(n– 1) distribution, the P-value for a

test of Ho against

  • These P-values are exact if the population distribution is normal and are approximately correct for large n in other cases.

Ha:μ > μo is P( T ≥ t)

Ha:μ < μo is P( T ≤ t)

Ha:μ≠ μo is 2P( T ≥|t|)

example 11 2 auto pollution
Example 11.2 - Auto Pollution
  • See example 11.2 on p.622

Minitab stemplot of the

data (page 623)

The one-sample t confidence interval has the form:

(where SE stands for “standard error”)