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The heights of pairs of fathers and their first sons are recorded.

It is claimed thatthere is no difference in heights between fathers and their first sons.

Corresponding to the given claim, the following hypotheses are set up:

Null Hypothesis: The mean of the differences between the height of fathers and their first sons is equal to 0.

\({H_0}:{\mu _d} = 0\)

Alternative Hypothesis: The mean of the differences between the height of fathers and their first sons is not equal to 0.

\({H_1}:{\mu _d} \ne 0\)

The test is two-tailed.

The following table shows the differences in the heights of the fathers and sons for each matched pair:

The number of pairs (n) is equal to 10.

The mean value of the differences is computed below:

\(\begin{array}{c}\bar d = \frac{{\left( { - 1} \right) + \left( { - 2} \right) + \ldots + 2}}{{10}}\\ = 0.02\end{array}\)

The standard deviation of the differences is computed below:

\(\begin{array}{c}{s_d} = \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({d_i} - \bar d)}^2}} }}{{n - 1}}} \\ = \sqrt {\frac{{{{\left( {\left( { - 1} \right) - 0.02} \right)}^2} + {{\left( {\left( { - 2} \right) - 0.02} \right)}^2} + ..... + {{\left( {2 - 0.02} \right)}^2}}}{{10 - 1}}} \\ = 1.86\end{array}\)

The mean value of the differences for the population of matched pairs \(\left( {{\mu _d}} \right)\) is considered to be equal to 0.

The value of the test statistic is computed as shown:

\(\begin{array}{c}t = \frac{{\bar d - {\mu _d}}}{{\frac{{{s_d}}}{{\sqrt n }}}}\\ = \frac{{0.02 - 0}}{{\frac{{1.86}}{{\sqrt {10} }}}}\\ = 0.034\end{array}\)

The degrees of freedom are computed below:

\(\begin{array}{c}df = n - 1\\ = 10 - 1\\ = 9\end{array}\)

Referring to the t-distribution table, the critical value of t at\(\alpha = 0.05\)and degrees of freedom equal to 9 for a two-tailed test are -2.2622 and 2.2622.

The right-tailed p-value of t equal to 0.034 is equal to 0.9729.

Since the p-value is greater than 0.05 and the test statistic lies between the two critical values, the null hypothesis is failed to reject.

There is insufficient evidence to support the claim that the height of fathers and their first sons have no difference.