Preliminary data analyses indicate that use of a paired...

Preliminary data analyses indicate that use of a paired t-test is reasonable. Perform each hypothesis test by using either the critical-value approach or the P-value approach. Sleep. In 1908. W. S. Gosset published "The Probable Error of a Mean" (Biometrika, Vol. 6, pp. $1-25$ ). In this pioneering paper, published under the pseudonym "Student," he introduced what later became known as Student's $t$ -distribution. Gossct used the following data set, which gives the additional slecp in hours obtained by 10 paticnts who used lacvohysocyamine hydrobromide. $$\begin{array}{|ccccc|} \hline 1.9 & 0.8 & 1.1 & 0.1 & -0.1 \\ 4.4 & 5.5 & 1.6 & 4.6 & 3.4 \\ \hline \end{array}$$ a. Identify the variable under consideration. b. Identify the two populations. c. Identify the paired-difference variable. d. Are the numbers in the table paired differences? Why or why not? e. At the $5 \%$ significance level, do the data provide sufficicnt evidence to conclude that laevohysocyamine hydrobromide is effective in increasing sleep? (Note: $\bar{d}=2.33$ and $s_{d}=2.002 .$ ) f. Repeat part (c) at the $1 \%$ significance level.

Preliminary data analyses indicate that use of a paired t-test is reasonable. Perform each hypothesis test by using either the critical-value approach or the P-value approach.
Sleep. In 1908. W. S. Gosset published "The Probable Error of a Mean" (Biometrika, Vol. 6, pp. $1-25$ ). In this pioneering paper, published under the pseudonym "Student," he introduced what later became known as Student's $t$ -distribution. Gossct used the following data set, which gives the additional slecp in hours obtained by 10 paticnts who used lacvohysocyamine hydrobromide.
$$\begin{array}{|ccccc|}
\hline 1.9 & 0.8 & 1.1 & 0.1 & -0.1 \\
4.4 & 5.5 & 1.6 & 4.6 & 3.4 \\
\hline
\end{array}$$
a. Identify the variable under consideration.
b. Identify the two populations.
c. Identify the paired-difference variable.
d. Are the numbers in the table paired differences? Why or why not?
e. At the $5 \%$ significance level, do the data provide sufficicnt evidence to conclude that laevohysocyamine hydrobromide is effective in increasing sleep? (Note: $\bar{d}=2.33$ and $s_{d}=2.002 .$ )
f. Repeat part (c) at the $1 \%$ significance level.

Key Concepts

Paired t-Test

A paired t-test is a statistical method used to compare the means of two related groups. It involves calculating the differences between paired observations (for example, measurements taken before and after an intervention) and then testing whether the mean of these differences is significantly different from zero. This test is appropriate when the data are naturally paired, and it accounts for the correlation between paired observations.

Hypothesis Testing

Hypothesis testing is a systematic method for making inferences about population parameters based on sample data. It involves stating a null hypothesis (typically, a statement of no effect or no difference) and an alternative hypothesis, then using statistical methods to decide whether there is sufficient evidence to reject the null hypothesis. In the context of this problem, the test determines if an intervention leads to a significant change in a measured response.

P-Value and Critical Value Approaches

The P-value approach and critical value approach are two methods of making a decision in hypothesis testing. The P-value approach involves calculating the probability of obtaining results as extreme as those observed, under the assumption that the null hypothesis is true, and comparing this probability to a predetermined significance level. The critical value approach involves comparing the test statistic to a threshold value from a statistical distribution; if the test statistic exceeds this threshold, the null hypothesis is rejected.

Paired Difference Variable

The paired difference variable is derived by taking the difference between paired observations from two related groups. This variable captures the change or effect that is of interest, such as the improvement or decline due to an intervention. Analyzing the distribution of these differences helps to determine whether there is a statistically significant mean change.

Significance Level

The significance level is a threshold value, commonly denoted as alpha, that sets the criterion for rejecting the null hypothesis. It represents the probability of committing a Type I error, which is the error of incorrectly rejecting a true null hypothesis. In hypothesis testing, if the calculated P-value is below the significance level, or if the test statistic exceeds the corresponding critical value, the null hypothesis is rejected.

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