Physiotherapy exam 2019-03-26

Degrees: Physiotherapy
Date: March 26, 2019

Question 1

The time required by a drug $A$ to be effective has been measured (in minutes) in a sample of 150 patients. The table below summarize the results.

$\begin{array}{lr} Response time & Patients \\ (0, 5] & 5 \\ (5, 10] & 15 \\ (10, 15] & 32 \\ (15, 20] & 36 \\ (20, 30] & 42 \\ (30, 60] & 20 \end{array}$

Are there outliers in the sample? Justify the answer.
What is the minimum time for the 20% of patients with highest response time?
What is the average response time? Is the mean representative?
Can we assume that the sample comes from a normal population?
If we take another sample of patients with mean 18 min and standard deviation 15 min, in which group is greater a response time of 25 min?

Use the following sums for the computations: $\sum x_{i} = 3105$ min, $\sum x_{i}^{2} = 83650$ min $^{2}$ , $\sum (x_{i} - \bar{x})^{3} = 206851.65$ min $^{3}$ y $\sum (x_{i} - \bar{x})^{4} = 8140374.96$ min $^{4}$ .

Solution

$Q_{1} = 12.7344$ min, $Q_{3} = 25.8333$ min, $I Q R = 13.099$ min, $f_{1} = - 6.9141$ min and $f_{2} = 45.4818$ min. Therefore there are outliers in the sample since the upper limit of the last interval is above the upper fence.
$P_{80} = 27.619$ min.
$\bar{x} = 20.7$ min, $s^{2} = 129.1767$ min $^{2}$ , $s = 11.3656$ min and $c v = 0.5491$ . The mean is moderately representative since the $c v \approx 0.5$ .
$g_{1} = 0.9393$ and $g_{2} = 0.2523$ . Since $g_{1}$ and $g_{2}$ are between -2 and 2, we can assume that the sample comes from a normal (bell-shaped) population.
The standard score of the first sample is $z (25) = 0.3783$ and the standard score of the second one is $z (25) = 0.4667$ , thus a time of 25 min is relatively greater in the second sample.

Question 2

In a regression study about the relation between two variables $X$ and $Y$ we got $\bar{x} = 7$ and $r^{2} = 0.9$ . If the equation of the regression line of $Y$ on $X$ is $y - x = 1$ , compute

The mean of $Y$ .
The equation of the regression line of $X$ on $Y$ .
What value does this regression model predict for $x = 6$ ? And for $y = 10$ ?

Solution

$\bar{y} = 8$ .
Regression line of $X$ on $Y$ : $x = 0.9 y - 0.2$ .
$y (6) = 7$ and $x (10) = 8.8$ .

Question 3

In a tennis club the age ( $X$ ) and the height ( $Y$ ) of the ten players conforming the female youth team has been measured.

$\begin{array}{lrrrrrrrrrr} Age (years) & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\ Height (cm) & 128 & 144 & 148 & 154 & 158 & 161 & 165 & 164 & 166 & 167 \end{array}$

Plot the scatter plot (Height on Age).
Which regression model bests fits these data, the linear or the logarithmic?
What is the expected height of a player 12.5 years old according to the best of two previous models?

Use the following sums for the computations:
$\sum x_{i} = 135$ years, $\sum \log (x_{i}) = 25.7908$ $\log (years)$ , $\sum y_{j} = 1555$ cm, $\sum \log (y_{j}) = 50.4358$ $\log (cm)$ ,
$\sum x_{i}^{2} = 1905$ years $^{2}$ , $\sum \log (x_{i})^{2} = 67.0001$ , $\log (years)^{2}$ , $\sum y_{j}^{2} = 243191$ cm $^{2}$ , $\sum \log (y_{j})^{2} = 254.4404$ $\log (cm)^{2}$ ,
$\sum x_{i} y_{j} = 21303$ years $\cdot$ cm, $\sum x_{i} \log (y_{j}) = 682.9473$ years $\cdot \log (cm)$ , $\sum \log (x_{i}) y_{j} = 4035.0697$ $\log (years)$ cm, $\sum \log (x_{i}) \log (y_{j}) = 130.2422$ $\log (years) \log (cm)$ .

Solution

2. $\bar{x} = 13.5$ years, $s_{x}^{2} = 8.25$ years $^{2}$ , $\overset{―}{\log (x)} = 2.5791$ log(years), $s_{\log (x)}^{2} = 0.0483$ log(years) $^{2}$ .
$\bar{y} = 155.5$ cm, $s_{y}^{2} = 138.85$ cm $^{2}$ . $s_{x y} = 31.05$ years $\cdot$ cm, $s_{\log (x) y} = 2.4594$ log(years)cm Linear coef. determination: $r^{2} = 0.8416$ Logarithmic coef. determination: $r^{2} = 0.9013$ Therefore, both models fit pretty well, but the logarithmic model fits a little bit better. 3. Logarithmic regression model: $y = 24.2639 + 50.8848 \log (x)$ . Prediction: $x (12.5) = 152.785$ cm.

Last updated on Sep 15, 2020