Statistics 7.3: Small-Sample Confidence Interval for a Population Mean (μ)

What happened when we replace the unknown σ with the value of the sample standard deviation s?

• estimating a doesn’t affect the center of our confidence intervals

• it does impact the proportion of intervals that contain the parameter by making our intervals too short

• this is because s values tend to be small in comparison to σ (the sampling distribution of s is right skewed)

• we will compensate by replacing the z critical value with a larger t critical value

t-distributions

are characterized by their degrees of freedom (df)

Degrees of freedom

number used in the denominator when calculate s, is the number of independent deviations available in the sample, for estimating σ

df = n - 1

Comparison between a standard normal distribution and t distribution

• both curves are similar in mound or bell shape, single peaked and symmetric about 0

• the relationship between mean and standard deviation seen in the normal distribution is not the same in t

• as the sample size and degrees of freedom increase, the t-distribution approaches the normal distribution

• the spread of the t-distribution is greater than the normal distribution (there is more area/probability)

Required Conditions

1. The sample was randomly selected from the population of interest (or there is some other indication)

2. The original population is known to be normal (or there is some other indication)

3. Population standard deviation (σ) is unknown

4. The sample size n is small