In \(1908\), W.S Gosset published "The Probable Error of Mean". In this pioneering paper, published under the pseudonym "Student" he introduced what later became known as Student's \(t-\)distribution, Gosset used the following data set, which gives the additional sleep in hours obtained by \(10\) patients who used laevohysocyamine hydrobromide.

a. Identify the variable under consideration.

b. Identify the two population.

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 sufficient evidence to conclude that laevohysocyamine hydrobromide is effective in increasing sleep?

f. Repeat part (e) at the \(1%\) significance level.

The values are \(\bar{d}=2.33, s_{d}=2.002\) and the data is shown below.

The variable is sleeping hours.

The two populations are sleep with and without laevohysocyamine hydrobromide.

The paired-difference variable is a pair of laevohysocyamine hydrobromide. One without laevohysocyamine hydrobromide and other is with laevohysocyamine hydrobromide.

The number in table are paired differences because the pair of laevohysocyamine hydrobromide. One without laevohysocyamine hydrobromide and other is with laevohysocyamine hydrobromide.

The null and alternative hypothesis are,

\(H_{0}:\mu_{1}=\mu_{2}\)

\(H_{a}:\mu_{1}>\mu_{2}\)

The test statistics is,

\(t=\frac{\bar{d}}{\frac{s_{d}}{\sqrt{n}}}\)

Substitute the given values in above equation.

\(t=\frac{2.33}{\frac{2.002}{\sqrt{10}}}\)

\(3.680\)

The degree of freedom is,

\(dof=n-1\)

\(=10-1\)

\(=9\)

The critical value for level of significance is \(1.833\).

Since, the value of test statistic is fall in the rejection region.

Thus, null hypothesis is rejected and it is reasonable to believe that the laevohysocyamine hydrobromide is effective in increasing sleep.

The critical value for level of significance is \(\pm 2.977\).

Since, the value of test statistic is fall in the rejection region.

Thus, null hypothesis is rejected and it is reasonable to believe that the laevohysocyamine hydrobromide is effective in increasing sleep.