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106  One Sample t-Test

106.1 Hypotheses -- Examples

Suppose we wish to test the following statistical hypothesis for a univariate, quantitative variable:

\[ \begin{cases}\text{H}_0: \mu = \mu_0 \\\text{H}_A: \mu \neq \mu_0\end{cases} \]

or

\[ \begin{cases}\text{H}_0: \mu \leq \mu_0 \\\text{H}_A: \mu > \mu_0\end{cases} \]

or

\[ \begin{cases}\text{H}_0: \mu \geq \mu_0 \\\text{H}_A: \mu < \mu_0\end{cases} \]

where \(\mu_0 = 50\). The chosen type I error \(\alpha\) is 5%.

The underlying theory is described in Chapter 96 (Statistical Test of the Population Mean with unknown Variance).

The corresponding test statistic is

\[ t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \sim t_{n-1} \quad (\text{under } H_0). \]

106.2 Analysis based on Critical Values

106.2.1 Software

The software can be found on the public website (https://compute.wessa.net/rwasp_hypothesismeanu.wasp) and on RFC (“Hypotheses / Empirical Tests”).

106.2.2 Data & Parameters

This R module on the public website contains the following fields:

  • Data (or variable X): a univariate dataset which represents quantitative data
  • Confidence: this is \(1 - \alpha\) (i.e. 1 minus the chosen type I error)
  • Null Hypothesis: this is the value of \(\mu_0\) against which the hypothesis is tested (in this case \(\mu_0 = 50\))

106.2.3 Output

Interactive Shiny app (click to load).
Open in new tab

Usually the confidence intervals are computed around the sample statistics (because the true population statistic is unknown). This can be seen in the two-sided interval: observe how the interval is symmetric around the sample mean 53.7, not around \(\mu_0 = 50\).

For one-sided inference, the direction (less or greater) must be chosen a priori from theory or study design (before observing the sample). Therefore, we report the two-sided interval by default and only use a one-sided interval when that direction was pre-specified.

The left-sided confidence interval contains the Null value \(\mu_0 = 50\). Therefore we fail to reject the Null Hypothesis which states that \(\mu \leq \mu_0 = 50\) (this assumes that we are testing a one-sided hypothesis).

The two-sided confidence interval also contains the Null value \(\mu_0 = 50\). Therefore we fail to reject the Null Hypothesis which states that \(\mu = \mu_0 = 50\) (assuming we are using a two-sided hypothesis test).

To compute the One Sample t-Test based on critical values on your local machine, the following script can be used in the R console.

Note: this local script uses a small illustrative vector. The embedded app example above uses a stored dataset and can produce different numeric output.

x <- c(50,48,44,56,61,52,53,55,67,51)
par1 = 0.95 #Confidence
par2 = 50 #Null Hypothesis
len <- length(x)
df <- len - 1
sd <- sd(x)
mx <- mean(x)
delta2 <- abs(qt((1-par1)/2,df)) * sd / sqrt(len)
delta1 <- abs(qt((1-par1),df)) * sd / sqrt(len)
#Sample Mean
print(mx)
#2-sided Confidence Interval
dum <- paste('[',mx-delta2)
dum <- paste(dum,',')
dum <- paste(dum,mx+delta2)
dum <- paste(dum,']')
print(dum)
#Left-sided Confidence Interval
dum <- paste('[',mx-delta1)
dum <- paste(dum,', +inf ]')
print(dum)
#Right-sided Confidence Interval
dum <- paste('[ -inf,',mx+delta1)
dum <- paste(dum,']')
print(dum)
[1] 53.7
[1] "[ 49.0024291302937 , 58.3975708697063 ]"
[1] "[ 49.8933776949979 , +inf ]"
[1] "[ -inf, 57.5066223050021 ]"

106.3 Analysis based on p-values

106.3.1 Software

The software can be found on the public website (https://compute.wessa.net/rwasp_onesampletests_mean.wasp) and on RFC (“Hypotheses / Empirical Tests”).

106.3.2 Data & Parameters

This R module on the public website contains the following fields:

  • Data: a univariate dataset which represents quantitative data
  • Alternative: parameter which defines the type of Hypothesis Test to be computed. This parameter can be set to the following values:
    • two.sided
    • less
    • greater
  • Confidence: this is \(1 - \alpha\) (i.e. 1 minus the chosen type I error)
  • Null Hypothesis: this is the value of \(\mu_0\) against which the hypothesis is tested (in this case \(\mu_0 = 50\))

106.3.3 Output

The p-value for the two-sided hypothesis is larger than the chosen type I error, i.e. \(p = 0.1085 > 0.05\). As a consequence we cannot reject the Null Hypothesis.

For reporting, also provide an effect size for the mean shift (Cohen 2013), e.g.

\[ d = \frac{\bar{x} - \mu_0}{s}. \]

When we specify Alternative = "greater" then the Alternative Hypothesis \(\text{H}_A: \mu > \mu_0\) is used.

This test corresponds to the test with the “left-sided” confidence interval that was discussed in the previous example. Since the p-value is larger than the chosen type I error, we cannot reject the Null Hypothesis. In the previous example we reached the same conclusion because the Null value was contained in the confidence interval.

The type I error \(\alpha\) must be fixed before the analysis. Changing \(\alpha\) after looking at the p-value is not valid inference.

Choosing Alternative = "less" corresponds to testing \(\text{H}_A: \mu < \mu_0\). This direction must be justified a priori (before looking at the sample). If the direction is selected post hoc, type I error control is lost.

To compute the One Sample t-Test based on critical values on your local machine, the following script can be used in the R console:

x <- c(50,48,44,56,61,52,53,55,67,51)
par1 = 'two.sided' #type of test
par2 = 0.95 #Confidence
par3 = 50 #Null Hypothesis
t.test(x,mu=par3,alternative=par1,conf.level=par2)

    One Sample t-test

data:  x
t = 1.7818, df = 9, p-value = 0.1085
alternative hypothesis: true mean is not equal to 50
95 percent confidence interval:
 49.00243 58.39757
sample estimates:
mean of x 
     53.7

106.4 Assumptions

The 95% confidence interval contains the true population mean in 95% of simple random samples that are (independently) drawn from the population. If we assume that the samples are (truly) random, we know that the sample mean is normally distributed if the number of observations is large enough (this is the case, regardless of the population distribution because of the Central Limit Theorem. If we are not sure about whether the sample is a genuine simple random sample then we have to explicitly make the assumption that the underlying population property is normal.

The number of sample observations is very small, i.e. \(N = 10\). This implies that we have to assume normality in the population and that we have to use the t-Distribution. When the sample is large enough, the distribution of the sample mean converges to normality. This, however, does not imply that we have to use another test (the so-called Z-Test). The reason is simple: the t-Test will always produce the correct answer because the t-Distribution converges to the Normal Distribution with increasing Degrees of Freedom.

106.5 Alternatives

There are several alternatives for the One Sample t-Test:

  • The Wilcoxon signed-rank test
  • Notched Boxplots
  • Bayesian tests
  • The Bootstrap Plot for Central Tendency

The confidence intervals for the mean are used in the following methods:

  • Trimmed Mean
  • Winsorized Mean