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.

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

  1. Are there outliers in the sample? Justify the answer.

  2. What is the minimum time for the 20% of patients with highest response time?

  3. What is the average response time? Is the mean representative?

  4. Can we assume that the sample comes from a normal population?

  5. 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$^4$.

  1. $Q_1=12.7344$ min, $Q_3=25.8333$ min, $IQR=13.099$ 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.
  2. $P_{80}=27.619$ min.
  3. $\bar x=20.7$ min, $s^2=129.1767$ min$^2$, $s=11.3656$ min and $cv=0.5491$. The mean is moderately representative since the $cv\approx 0.5$.
  4. $g_1=0.9393$ 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.
  5. 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

  1. The mean of $Y$.

  2. The equation of the regression line of $X$ on $Y$.

  3. What value does this regression model predict for $x=6$? And for $y=10$?

  1. $\bar y=8$.
  2. Regression line of $X$ on $Y$: $x=0.9y-0.2$.
  3. $y(6)=7$ 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.

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

  1. Plot the scatter plot (Height on Age).

  2. Which regression model bests fits these data, the linear or the logarithmic?

  3. 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, $\sum \log(x_i)=25.7908$ $\log(\mbox{years})$, $\sum y_j=1555$ cm, $\sum \log(y_j)=50.4358$ $\log(\mbox{cm})$,
$\sum x_i^2=1905$ years$^2$, $\sum \log(x_i)^2=67.0001$, $\log(\mbox{years})^2$, $\sum y_j^2=243191$ cm$^2$, $\sum \log(y_j)^2=254.4404$ $\log(\mbox{cm})^2$,
$\sum x_iy_j=21303$ years$\cdot$cm, $\sum x_i\log(y_j)=682.9473$ years$\cdot\log(\mbox{cm})$, $\sum \log(x_i)y_j=4035.0697$ $\log(\mbox{years})$cm, $\sum \log(x_i)\log(y_j)=130.2422$ $\log(\mbox{years})\log(\mbox{cm})$.

  1. Scatter plot of Height on Age

2.$\bar x=13.5$ years, $s_x^2=8.25$ years$^2$, $\overline{\log(x)}=2.5791$ log(years), $s_{\log(x)}^2=0.0483$ log(years)$^2$.
$\bar y=155.5$ cm, $s_y^2=138.85$ cm$^2$. $s_{xy}=31.05$ 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.

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Last updated on Sep 15, 2020

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