Question

A manufacturer has recorded its cost of electricity (cost) and the total number of hours of machine use time (time) each week for a total of 52 weeks. Use the Mod7-1Data (Links to an external site.) to answer the following questions. USE α = 5%!

Time	Cost
7	886.44
14	1598.05
15	1385.82
8	594.93
8	792.09
15	1307.7
13	1334.7
12	900.14
11	1306.41
11	950.22
13	1249.98
14	1388.16
6	750.87
14	1185.85
8	683.89
12	1245.1
14	1342.55
15	1154.05
11	940.88
14	992.5
8	886.89
13	980.66
14	1171.71
10	1185.57
11	987.64
11	1035.9
11	902.04
11	1195.62
8	1001.72
13	1421.23
9	684.21
8	497.48
12	1043.28
15	1054.48
11	1043.03
9	876.73
6	833.68
8	815.6
15	1396.96
11	1103.17
7	464.26
12	1159.15
9	959.89
11	1171.85
8	740.38
7	1001.72
11	865.59
13	1237.42
9	1295.29
12	1527.08
10	962
7	688.7

sample covariance=

sample correlation=

null hypothesis=

computed test statistic=

alternative hypothesis=

statistical conclusion=

Answer 1

1. For the given data, the covariance of Time and cost can be computed using the formula:

Using excel,

We get Covariance between Time and Cost = 498.902

2. For the given data, the linear relation between of Time and cost (two continuous variable) can be computed using Pearson's Correlation (r) obtained using the formula:

Using excel,

We get r = 0.737

3. Now, we have to test whether this correlation coefficient is significant at 5% level. Le denote the population correlation coefficient.We have to test:

Null hypothesis: Vs

Alternative hypothesis: (Two sided test)

The appropriate statistical test to test the above hypothesis would be a t test, with test statistic given by:

Computing the test statistic:

We get t statistic = 7.718

Computing the p-value: (We would determine whether the p-value is less than the significance level, if yes, the null hypothesis is rejected)

Since, p-value = 0.000 < 0.05, we may reject H0 at 5% level. We may conclude that there exists a significant relationship between  Time and Cost. Also, from the sample correlation coefficidnt, we find that the significant relationship is strong and positive.