Question

The degenerative disease osteoarthritis most frequently affects weight-bearing joints such as the knee. The article "Evidence of Mechanical Load Redistribution at the Knee Joint in the Elderly When Ascending Stairs and Ramps" (Annals of Biomed. Engr., \(2008: 467 - 476\)) presented the following summary data on stance duration (ms) for samples of both older and younger adults.

\(\begin{array}{*{20}{l}}{Age\;\;\;\;\;\;\;Sample Size\;\;Sample Mean\;\;Sample SD}\\{\;Older\;\;\;\;\;28\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;801\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;117}\\{Younger\;16\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;780\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;72}\end{array}\)

Assume that both stance duration distributions are normal.

a. Calculate and interpret a\(99\% \)CI for true average stance duration among elderly individuals.

b. Carry out a test of hypotheses at significance level\(.05\)to decide whether true average stance duration is larger among elderly individuals than among younger individuals.

Short answer

(a) \(( - 56.0945,98.0945)\)

(b) There is not sufficient evidence to support the claim that the true average stance duration is larger among elderly individuals than among younger individuals.

Step-by-step solution

a)Step 1: Find the end point of confidence interval

\(\begin{array}{l}{{\bar x}_1} = 801\\{{\bar x}_2} = 780\\{n_1} = 28\\{n_2} = 16\\{s_1} = 117\\{s_2} = 72\\c = 99\% = 0.99\\\alpha = 0.05\end{array}\)

Determine the degrees of freedom (rounded down to the nearest integer):

\(\Delta = \frac{{{{\left( {\frac{{s_1^2}}{{{n_1}}} + \frac{{s_2^2}}{{{n_2}}}} \right)}^2}}}{{\frac{{{{\left( {s_1^2/{n_1}} \right)}^2}}}{{{n_1} - 1}} + \frac{{{{\left( {s_2^2/{n_2}} \right)}^2}}}{{{n_2} - 1}}}} = \frac{{{{\left( {\frac{{{{117}^2}}}{{28}} + \frac{{{{72}^2}}}{{16}}} \right)}^2}}}{{\frac{{{{\left( {{{117}^2}/28} \right)}^2}}}{{28 - 1}} + \frac{{{{\left( {{{72}^2}/16} \right)}^2}}}{{16 - 1}}}} \approx 41 > 40\)

Determine the t-value by looking in the row starting with degrees of freedom \(df = 40\) and in the column with\(1 - c/2 = 0.005\) in the Student's distribution table in the appendix:

\({t_{\alpha /2}} = 2.704\)

The margin of error is then:

\(E = {t_{\alpha /2}} \cdot \sqrt {\frac{{s_1^2}}{{{n_1}}} + \frac{{s_2^2}}{{{n_2}}}} = 2.704 \cdot \sqrt {\frac{{{{117}^2}}}{{28}} + \frac{{{{72}^2}}}{{16}}} \approx 77.0945\)

The endpoints of the confidence interval for\({\mu _1} - {\mu _2}\)are:

\(\begin{array}{l}\left( {{{\bar x}_1} - {{\bar x}_2}} \right) - E = (801 - 780) - 77.0945 = 21 - 77.0945 = - 56.0945\\\left( {{{\bar x}_1} - {{\bar x}_2}} \right) + E = (801 - 780) + 77.0945 = 21 + 77.0945 = 98.0945\end{array}\)

b)Step 2: Determine the test statistic

(b) Given claim: larger

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis and the alternative hypothesis state the opposite of each other. The null hypothesis needs to contain the value mentioned in the claim.

\(\begin{array}{l}{H_0}:{\mu _1} = {\mu _2}\\{H_a}:{\mu _1} > {\mu _2}\end{array}\)

Determine the test statistic:

\(t = \frac{{{{\bar x}_1} - {{\bar x}_2}}}{{\sqrt {\frac{{s_1^2}}{{{n_1}}} + \frac{{s_2^2}}{{{n_2}}}} }} = \frac{{801 - 780}}{{\sqrt {\frac{{{{117}^2}}}{{28}} + \frac{{{{72}^2}}}{{16}}} }} \approx 0.737\)

The\({\rm{P}}\)-value is the probability of obtaining the value of the test statistic, or a value more extreme. The\({\rm{P}}\)-value is the number (or interval) in the column title of Student's T distribution in the appendix containing the t-value in the row\(df = 40\):

\(P > 0.10\)

If the P-value is less than or equal to the significance level, then the null hypothesis is rejected:

\(P > 0.05 \Rightarrow {\rm{ Fail to reject }}{H_0}\)

There is not sufficient evidence to support the claim that the true average stance duration is larger among elderly individuals than among younger individuals.