a marketing company conducted a survey to assess the audience response to different aspects of a new…

Explanation:

Step1: Recall correlation - coefficient formula

The formula for the correlation coefficient $r$ is $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$, where $n$ is the number of data - points, $x$ is the duration of the ad, and $y$ is the number of viewers who disliked the ad. Let $x$ be the duration of the ad (in seconds) and $y$ be the number of viewers who disliked the ad. First, calculate the following sums for $n = 12$ data - points:

$x$	$y$	$xy$	$x^{2}$	$y^{2}$
18	24	432	324	576
22	32	704	484	1024
26	30	780	676	900
30	41	1230	900	1681
34	32	1088	1156	1024
38	45	1710	1444	2025
42	31	1302	1764	961
46	38	1748	2116	1444
50	41	2050	2500	1681
54	52	2808	2916	2704
60	62	3720	3600	3844
$\sum x=420$	$\sum y = 438$	$\sum xy=17672$	$\sum x^{2}=18880$	$\sum y^{2}=18865$

Step2: Calculate the numerator

$n(\sum xy)-(\sum x)(\sum y)=12\times17672 - 420\times438$ $=212064-183960 = 28104$

Step3: Calculate the first part of the denominator

$n\sum x^{2}-(\sum x)^{2}=12\times18880-420^{2}$ $=226560 - 176400=50160$

Step4: Calculate the second part of the denominator

$n\sum y^{2}-(\sum y)^{2}=12\times18865 - 438^{2}$ $=226380-191844 = 34536$

Step5: Calculate the denominator

$\sqrt{(n\sum x^{2}-(\sum x)^{2})(n\sum y^{2}-(\sum y)^{2})}=\sqrt{50160\times34536}$ $=\sqrt{1732209760}\approx41617.42$

Step6: Calculate the correlation coefficient

$r=\frac{28104}{41617.42}\approx0.675$

Since the correlation coefficient $r\approx0.675$ which is positive, we can conclude that there is a positive linear relationship between the duration of advertisements and the audience disliking them.

Answer:

The correlation coefficient for this data set is close to $0.675$. Based on this information, we can conclude that there is a positive linear relationship between the duration of advertisements and the audience disliking them.