1. [20 points] For each of the statements below, circle T or F for ``True'' or ``False,'' respectively. (4 pts. each)

: The correlation can be any number, but it is usually between -1 and +1. FALSE: Correlation must be between -1 and +1. See item 1. at the bottom of p. 166 in the text.

F : If the distribution is bell shaped with no outliers, we expect IQR will be smaller than s. FALSE: It was stated in class that IQR is approximately equal to 1.35 s for the normal distribution. You can also figure this out from the tables.

F : Nonresponse bias refers to systematic error in sampling from a population due to subjects being unavailable or refusing to reply. TRUE: See bottom of p. 251 in the text.

F : One purpose of randomization in experimental design is to eliminate confounding effects from lurking variables that might be present in an observational study. TRUE: This was stated in lecture.

F : Both the mean and the median are resistant measures of the center of a distribution of data. FALSE: See the bottom of p. 37 to the top of p. 38.

2. [30 points] Use the table below to sketch a density histogram for the data.

Class 1c|Percentage 1c|Class 1c|Bar

1c|Width 1c|Height

$$\longrightarrow$$ 0 - 20 10% 20 0.5

$$\longrightarrow$$ 20 - 30 20% 10 2.0

$$\longrightarrow$$ 30 - 40 30% 10 3.0

$$\longrightarrow$$ 40 - 60 20% 20 1.0

$$\longrightarrow$$ 60 - 100 20% 40 0.5

We have added two extra columns in the table : one for class width and one for the height of the histogram bars. The plot appears below.

3. [30 points] Suppose a data set has approximately a normal distribution with mean $\bar{x}$ = 200 and standard deviation s = 20.

(a) Estimate the percentage of the data which are between 170 and 210.

Computing the corresponding z values:

From the tables provided, the area under the curve to the left of z₁ = -1.5 is 0.0668, and the area under the curve to the left of z₂ = 0.5 is 0.6915. Thus, the area between them is

$$0.6915 \, - \, 0.0668 \; = \; 0.6247 .$$

The calculation is depicted pictorially below.

3(b) Find approximately the 80'th percentile of the data.

Using the tables, the 80'th percentile of the N(0,1) distribution is 0.84. The area corresponding the z = 0.84 is 0.7995, which is the closest we can get to 0.8. The corresponding data value is obtained by the ``inverse'' z-value transformation:

$$x \; = \; \bar{x} \, + \, z * s \; = \; 200 \, + \, 0.84*20 \; = \; 216.8 .$$

Thus, our estimate of the 80'th percentile of the data is 216.8.

4. [20 points] Below are 5 values of r, the correlation of a sample, and 4 scatterplots. Match the value of r with the scatterplot by writing the plot label (A, B, C, or D) next to the value of r. Obviously, one value of r will be unmatched.

(i)

r = -0.9 Plot C. Clearly this plot has a negative association and the points fall close to a straight line, so the corresponding correlation is near -1

(ii)

r = -0.5 No Plot.

(iii)

r = 0.0 Plot B. There is no upward or downward ``tilt'' in Plot B, so there is no linear association, although there is a clear nonlinear association.

(iv)

r = +0.5 Plot A. Plot A displays a fairly clear upward tilt, hence has a positive correlation, but it is not so strongly positive as Plot D, so by process of elimination Plot A must go with r = +0.5.

(v)

r = +0.9 Plot D. Plot D has a very strong positive correlation, near 1.

Summary Statistics for Scores

For n = 75 persons taking the exam before Fri., 13 Feb.:

$$\bar{x} \; = \; 87.01 , \quad s \; = \; 13.07 .$$

Five Number summary:

$$\begin{array} {ccccc} \mbox{min} & Q_1 & M & Q_3 & \mbox{max} \\ 56 & 77 & 92 & 99 & 100\end{array}$$

Histogram is shown below. Approximate letter grades are

$$\begin{array} {ccrr} \mbox{A-B} & & 85--100 & (68\%) \\ \mbox{C} & & 65-84 & (25\%) \\ \mbox{D} & & 56-64 & (7\%)\end{array}$$