9.2 - Estimating a Population Mean

Recall:

  • The confidence interval again has the form

    • point estimate ± margin of error.

  • The sample mean is the point estimate of the

    • population mean.

  • The exact standard deviation of the sample mean is called the

    • Standard error

The t Distribution and its Properties:

If we know the standard error (aka the SD of the sample mean), then under the circumstances in which the population is normal, then we can use the formula…

  • For instance, with z = 1.96 for 95% confidence. This interval would contain the population mean 95% of the time.

  • To account for this increased error, we must replace the z-score by a slightly larger score, called a t-score. The confidence interval is then a bit wider— this distribution is called the

    t distribution.

    • The t distribution resembles the standard normal distribution, being bell-shaped around a mean of 0. 

    • Its standard deviation is a bit larger than 1, the precise value depending on what is called the degrees of freedom, denoted by df. 

    • For inference about a population mean, the degrees of freedom equal df = n – 1, one less than the sample size.

    • Summary: Properties of the t Distribution

      • The t distribution is bell-shaped and symmetric about 0.

      • The probabilities depend on the degrees of freedom, df = n – 1

      • The t distribution has thicker tails than the standard normal distribution, i.e., it is more spread out.

      • A t -score multiplied by the standard error gives the margin of error for a confidence interval for the mean.

Figure 8.7 The t Distribution Relative to the Standard Normal Distribution. The t distribution is more spread out than the standard normal but gets closer to it as the degrees of freedom (df) increase. The two are practically identical when df ≥ 30. Question Can you find z-scores (such as 1.96) for a normal distribution on the t table (Table B)?

Table 8.3 Part of Table B Displaying t-Scores. The scores have right-tail probabilities of 0.100, 0.050, 0.025, 0.010, 0.005, and 0.001. When n = 7, df = 6, and t.025 = 2.447 is the t -score with right-tail probability = 0.025 and two-tail probability = 0.05. It is used in a 95% confidence interval, x-bar ± 2.447 (se)