Question

Movie companies need to predict the gross receipts of individual movies after a movie has debuted. The accompanying results are the first weekend​ gross, the national​ gross, and the worldwide gross​ (in millions of​ dollars) of six movies. Complete parts​ (a) through​ (d) below..

a. Compute the covariance between first weekend gross and national​ gross, first weekend gross and worldwide​ gross, and national gross and worldwide gross.

QUESTION 1: Find the covariance between first weekend gross and national gross. nothing ​(Round to four decimal places as​ needed.)

Title First Weekend   National Gross   Wordwide Gross
Movie A 90.868 317.036 976.867
Movie B 88.947 261.469 878.796
Movie C 93.682 249.394 795.296
Movie D 102.506 290.303 896.347
Movie E 77.133 292.853 938.099
Movie F 77.965 301.232 934.742

Answer 1

We will calculate the sample covariances, as the data we have is a sample of 6 movies.

Sample covariance between 2 variables X and Y is calculated as (we will use n-1 as the denominator, as against "n" in case of population covariance)

$Cov(X,Y)=\frac{1}{n-1}\sum(x-\bar{x})(y-\bar{y})$

where n is the number of observations

$\bar{x}=\frac{\sum x}{n}$ is the sample mean of X

$\bar{y}=\frac{\sum y}{n}$ is the sample mean of Y

Let X be the First Weekend gross

Y be the National gross

Z be the worldwide gross

The sample means are

$\bar{x}=\frac{\sum x}{n}=\frac{90.868+ 88.947+ 93.682+ 102.506+ 77.133+ 77.965}{6}=88.517$
$\begin{align*} \bar{y}=\frac{\sum y}{n}=\frac{317.036+ 261.469+ 249.394+ 290.303+ 292.853+ 301.232}{6}=285.381 \end{align*}$

$\begin{align*} \bar{z}=\frac{\sum z}{n}=\frac{976.867+ 878.796+ 795.296+ 896.347+ 938.099+ 934.742}{6}=903.358 \end{align*}$

a) the sample covariance between first weekend gross (X) and national​ gross (Y) is

$\begin{align*} Cov(X,Y)&=\frac{1}{n-1}\sum(x-\bar{x})(y-\bar{y})\\ &=\frac{1}{6-1}\left((90.868-88.517)\times (317.036-285.281)+ (88.947-88.517)\times (261.469-285.281)+ (93.682-88.517)\times (249.394-285.281)+ (102.506-88.517)\times (290.303-285.281)+ (77.133-88.517)\times (292.853-285.281)+ (77.965-88.517)\times (301.232-285.281) \right )\\ &=-61.0403 \end{align*}$

ans: The sample covariance between first weekend gross and national​ gross is -61.0403

the sample covariance between first weekend gross (X) and worldwide​ gross (Z) is

$\begin{align*} Cov(X,Z)&=\frac{1}{n-1}\sum(x-\bar{x})(z-\bar{z})\\ &=\frac{1}{6-1}\left((90.868-88.517)\times (976.867-903.358)+ (88.947-88.517)\times (878.796-903.358)+ (93.682-88.517)\times (795.296-903.358)+ (102.506-88.517)\times (896.347-903.358)+ (77.133-88.517)\times (938.099-903.358)+ (77.965-88.517)\times (934.742-903.358) \right )\\ &=-244.1229 \end{align*}$

ans: the sample covariance between first weekend gross and worldwide​ gross is -244.1229

The sample covariance between national gross (Y) and worldwide gross (Z) is

$\begin{align*} Cov(Y,Z)&=\frac{1}{n-1}\sum(y-\bar{y})(z-\bar{z})\\ &=\frac{1}{6-1}\left((317.036-285.381)\times (976.867-903.358)+ (261.469-285.381)\times (878.796-903.358)+ (249.394-285.381)\times (795.296-903.358)+ (290.303-285.381)\times (896.347-903.358)+ (292.853-285.381)\times (938.099-903.358)+ (301.232-285.381)\times (934.742-903.358)\right )\\ &=1505.1251 \end{align*}$

ans: The sample covariance between national gross and worldwide gross is 1505.1251