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Section 7.4 Summary of One-sample \(t\) Procedures

One-sample \(t\) Procedures.

Parameter: \(\mu\) = the population mean
To test \(H_0: \mu = \mu_0\)
Standardized (test) statistic:
\begin{equation*} t_0 = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \end{equation*}
Degrees of freedom = \(n - 1\)
\(t\)-Confidence interval for \(\mu\)
\begin{equation*} \bar{x} \pm t_{n-1}^* \frac{s}{\sqrt{n}} \end{equation*}
Technical conditions: These procedures are considered valid if the sample distribution is reasonably symmetric or the sample size is at least 30.

Technology Instructions.

Hint 1. R
Raw data:
t.test(data, mu = hypothesized, alt = "greater", "less", or "two.sided", conf.level)
Summary data:
iscamonesamplet(xbar, sd, n, hypothesized, alternative, conf.level)
Hint 2. JMP
Raw data: Analyze > Distribution
Summary data: Use ISCAM Journal file: Hypothesis Test for One Mean and Confidence Interval for One Mean. You can specify a column of data (raw data) or summary statistics (summary data); select the \(t\)-test and \(t\) interval radio buttons.
Hint 3. Theory-Based Inference applet
Use the pull-down menu to select One Mean. Specify the summary statistics (the sample size, sample mean, and sample standard deviation \(s\)) or paste in the raw data. Check the box for test of significance and specify the hypothesized value, use the \(\neq\) button to specify the direction of the alternative (\(<\) \(>\) for not equal) and press Calculate and/or check the box for confidence interval, specify the confidence level and press Calculate CI. The applet can report both the confidence interval and the prediction interval.

Prediction Intervals for Individual Observations.

Prediction Interval:
\begin{equation*} \bar{x} \pm t_{n-1}^* s\sqrt{1+\frac{1}{n}} \end{equation*}
Valid only with normal population of observations

Technology Instructions.

Hint 1. R
predict(lm(y ~ 1), newdata = data.frame(var = 0), interval = "predict")
Hint 2. JMP
Raw data: From Analyze > Distribution, use the hot spot to select Prediction Interval
Hint 3. Theory-Based Inference applet
Check the box for the Prediction interval
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