Question

In: Statistics and Probability

An emergency service wishes to see whether a relationship exists between the outside temperature and the...

An emergency service wishes to see whether a relationship exists between the outside temperature and the number of emergency calls it receives for a 7-hour period. The data are shown.

Temperature x :68 74 82 88 93 99 101

No. of calls y : 7 4 8 10 11 9 13

a) Draw the scatter plot for the variables.

b) Compute and explain the correlation coefficient in terms of the question provided.

c) Test the significance of the correlation coefficient at ? =0.05. Show all steps of

the hypothesis test.

d) What percent of the variability is accounted for by the relationship between the two

variables?

e) What would be the slope and y-intercept for a regression line based on this data?

Write the regression line also.

f) Draw the regression line on the scatterplot from part a.

g) What is the coefficient of non-determination?

h) Find y’ when x =80 F.

i) Run the regression analysis in R using the lm function and report the R output.

Solutions

Expert Solution

a)

Following is the scatter plot:

b)

Following table shows the calculations:

X Y X^2 Y^2 XY
68 7 4624 49 476
74 4 5476 16 296
82 8 6724 64 656
88 10 7744 100 880
93 11 8649 121 1023
99 9 9801 81 891
101 13 10201 169 1313
Total 605 62 53219 600 5535

Sample size: n=7

Now,

The coefficient of correlation is :

c)

Hypotheses are:

Test statistics:

-------------------------

Here degree of freedom is df=n-2=5 so p-value of the test is 0.0267.
The excel function used for p-value is: "=TDIST(3.104,5,2)"

Since p-value is less than 0.05 so we reject the null hypothesis.

d)

The percent of the variability is accounted for by the relationship between the two variables is

e)

Slope of the regression equation is

and intercept of the equation will be

So the regression equation will be

y'=-7.5441+0.1898x

f)

g)

The coefficient of non-determination is

h)

y'=-7.5441+0.1898*80 = 7.6399

i)

Following is the screen shot of R script:

Following is the R output:


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