Inference for One Mean

Elementary Statistics

MTH-161D | Spring 2025 | University of Portland

March 31, 2025

Objectives

These slides are derived from Diez et al. (2012).

Previously…

Central Limit Theorem (CLT)

CLT says that the sample mean (or sum) of many independent and identically distributed random variables approaches a normal distribution, regardless of the original distribution.

CLT Conditions

Central Limit Theorem for the Sample Mean

When we collect a sufficiently large sample of \(n\) independent observations from a population with mean \(\mu\) and standard deviation \(\sigma,\) the sampling distribution of \(\bar{x}\) will be nearly normal with \[\text{Mean} \longrightarrow \mu \text{ and } \text{Standard Error} \longrightarrow SE = \frac{\sigma}{\sqrt{n}}.\]

Evaluating the two conditions required for modeling \(\bar{x}\)

Two conditions are required to apply the Central Limit Theorem for a sample mean \(\bar{x}:\)

General rule for performing the normality check

Note, it often takes practice to get a sense for whether or not a normal approximation is appropriate.

Normality Assesment (1/2)

Consider the four plots provided that come from simple random samples from different populations.

Are the independence and normality conditions met in each case?

Histograms of samples from two different populations.

Histograms of samples from two different populations.

Normality Assesment (2/2)

The t-distribution (1/2)

Comparison of a $t$-distribution and a normal distribution.

Comparison of a \(t\)-distribution and a normal distribution.

The \(t\)-distribution is always centered at zero and has a single parameter: degrees of freedom. The degrees of freedom describes the precise form of the bell-shaped \(t\)-distribution. In general, we’ll use a \(t\)-distribution with \(df = n - 1\) to model the sample mean when the sample size is \(n\).

The t-distribution (2/2)

The larger the degrees of freedom the more closely the $t$-distribution resembles the standard normal distribution.

The larger the degrees of freedom the more closely the \(t\)-distribution resembles the standard normal distribution.

Case Study I: Mercury content in Risso’s dolphins

We will identify a confidence interval for the average mercury content in dolphin muscle using a sample of 19 Risso’s dolphins from the Taiji area in Japan.

Summary of mercury content in the muscle of 19 Risso’s dolphins from the Taiji area. Measurements are in micrograms of mercury per wet gram of muscle \((\mu\)g/wet g).
n Mean SD Min Max
19 4.4 2.3 1.7 9.2

Are the independence and normality conditions satisfied for this dataset?

Case Study I: One-sample t-interval (1/2)

One sample t-intervals

\[ \begin{aligned} \text{point estimate} \ &\pm\ t^*_{df} \times SE \\ \bar{x} \ &\pm\ t^*_{df} \times \frac{s}{\sqrt{n}} \end{aligned} \]

Using R to find \(t^*\)

qt(0.025, df = 18)
## [1] -2.100922

Case Study I: One-sample t-interval (2/2)

One sample t-intervals

\[ \begin{aligned} \bar{x} \ &\pm\ t^*_{18} \times SE \\ 4.4 \ &\pm\ 2.10 \times 0.528 \\ \end{aligned} \] \[(3.29,5.51)\]

We are 95% confident the average mercury content of muscles in Risso’s dolphins is between 3.29 and 5.51 \(\mu\)g/wet gram, which is considered extremely high.

Activity: Determine Confidence Intervals for One Mean

  1. Make sure you have a copy of the M 3/31 Worksheet. This will be handed out physically. This worksheet will be available on Moodle after class.
  2. Work on your worksheet by yourself for 10 minutes. Please read the instructions carefully. Ask questions if anything need clarifications.
  3. Get together with another student.
  4. Discuss your results.
  5. Submit your worksheet on Moodle as a .pdf file.

References

Diez, D. M., Barr, C. D., & Çetinkaya-Rundel, M. (2012). OpenIntro statistics (4th ed.). OpenIntro. https://www.openintro.org/book/os/