Chapter 1, Problem 8P

Noise and Intelligibility Audiologists study the intelligibility of spoken sentences un­der different noise levels. Intelligibility, the MRT score, is measured as the percent of a spoken sentence that the listener can decipher at a certain noise level in decibels (dB). The table shows the results of one such test.

• (a) Make a scatter plot of the data.
• (b) Find and graph the regression line.
• (c) Find the correlation coefficient. Is a linear model appropriate?
• (d) Use the linear model in part (b) to estimate the intelligibility of a sentence at a 94-dB noise level.

Noise level (dB)	MRT score (%)
80	99
84	91
88	84
92	70
96	47
100	23
104	11

To determine

To graph: The scatter plot of the data.

Explanation of Solution

Graph:

Consider the Noise level in decibel as the x coordinates and the M.R.T score as the y coordinates.

The scatter plots of the given data are shown below in Figure 1.

From Figure 1, all the points are plotted on the graph. No two points coincide each other.

To determine

To find: The regression line that represents the given data and draw the graph for it.

Answer to Problem 8P

The regression that represents the given data is $y=-3.902x+419.7$.

Explanation of Solution

By the use of online calculator, the regression line for the given data obtained is $y=-3.902x+419.7$, where M.R.T score is the y-intercept and noise level is the x-intercept.

The above regression line is in the form of linear equation.

Therefore, the linear function that represents the given data is $y=-3.902x+419.7$.

The graph that represents the equation $y=-3.902x+419.7$ is shown below in Figure 1.

From Figure 2, the graph for the linear equation $y=-3.902x+419.7$ is a straight line.

To determine

To find: The correlation coefficient and to check whether the linear model appropriate or not.

Answer to Problem 8P

The correlation coefficient is $-0.76$.

Explanation of Solution

Formula used:

The formula for the correlation coefficient is $r=\frac{n\left(\sum xy\right)-\left(\sum x\right)\left(\sum y\right)}{\sqrt{\left(n\sum x^2-{\left(\sum x\right)}^2\right)\left(n\sum y^2-{\left(\sum y\right)}^2\right)}}$.

Calculation:

Compute the values of the summation of x, y, xy, $x^2$, $y^2$ and are shown in below table.

x	y	xy	$x^2$	$y^2$
−10	99	−990	100	9801
−6	91	−546	36	8281
−2	84	−168	4	7056
2	70	140	4	4900
6	47	282	36	2209
10	23	230	100	529
14	11	154	196	121
$\sum x$=8	$\sum y$=425	$\sum xy$=−898	$\sum x^2$=476	$\sum y^2$=32897

It is observed that the value of n = 7.

Substitute the values that are obtained in the table to compute the correlation coefficient by using the above mentioned formula.

$\begin{array}{c}r=\frac{7\left(-898\right)-\left(8\right)\left(425\right)}{\sqrt{\left(7\left(476\right)-{\left(8\right)}^2\right)\left(7\left(32897\right)-{\left(425\right)}^2\right)}}\\ =\frac{-6289-3400}{\sqrt{\left(3332-64\right)\left(230279-180625\right)}}\\ =\frac{-9689}{\sqrt{\left(3268\right)\left(49654\right)}}\\ =\frac{-9689}{\sqrt{162269272}}\end{array}$

On further simplification.

$\begin{array}{c}r=\frac{-9689}{12738.49567}\\ =-0.76\end{array}$

Thus, the value of the correlation coefficient is $-0.76$.

To determine

To find: The estimate value of the intelligibility of a sentence at a 94-dB.

Answer to Problem 8P

The estimate value of the intelligibility of a sentence at a 94-dB is 52.912 %.

Explanation of Solution

Consider the noise level at 94-dB.

From part (b), it is obtained that the linear equation that represents the given data is $y=-3.902x+419.7$.

Substitute x = 94 in the above equation to compute the M.R.T score.

$\begin{array}{c}y=-3.902\left(94\right)+419.7\\ =-366.788+419.7\\ =52.912\end{array}$

Thus, the estimate value of the intelligibility of a sentence at a 94-dB is 52.912 %.