Question

s Question Help Media periodically discuss the issue of heights of winning presidential candidates and heights of their main opponents. The accompanying table lists the heights (cm) from several recent presidential elections. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value ofr. Determine whether there is sufficient evidence to support a claim of linear correlation between the two variables. Should we expect that there would be a correlation? Use a significance level of a = 0.05. Click the icon to view the heights of the candidates. Opponen Opponen 160+ 160 200 President Height (cm) 160+ 160 200 President Height (cm) 1601 160 200 President Height (cm) 160+ 160 200 President Height (cm) The linear correlation coefficient ris I (Round to three decimal places as needed.) Х Candidate Heights Determine the null and alternative hypotheses. Hop H:P V (Type integers or decimals. Do not round.) President 183 179 193 174 183 188 193 186 181 188 184 184 178 184 Opponent 185 180 185 178 173 184 181 175 185 175 177 190 183 174 The test statistic is (Round to two decimal places as needed.) Print Done The P-value is (Round to three decimal places as needed.) Because the P-value of the linear correlation coefficient is the significance level, there sufficient evidence to support the claim that there is a linear correlation between the heights of winning presidential candiates and the heights Click to select your answer(s). ?

Answer 1

Ans Given data set are below

President Height cm (x)	Opponent height cm (y)
183	185
179	180
193	185
174	178
183	173
188	184
193	181
186	175
181	185
188	175
184	177
184	190
178	183
184	174

To Construct Scatter plot for this data are

192 190 188 186 M 184 M Opponent Height (v) 182 . Opponent Height (y) 180 178 176 174 IT 172 170 175 190 195 180 185 President Height (x)

Comment : There are no relation between them

Now to calculate linear correlation coefficient using following formula :

Σ(x^2 – Mx^2)(y - My) r – Σ(x^2 – Mx^2)* * Σy – My)?

President Height (x)	Opponent Height (y)	(X-Mx)*(y-My)	(X-Mx)^2	(Y-My)^2
183	185	-5.30612	1.306122	21.55612
179	180	1.836735	26.44898	0.127551
193	185	41.12245	78.44898	21.55612
174	178	23.90816	102.8776	5.556122
183	173	8.408163	1.306122	54.12755
188	184	14.05102	14.87755	13.27041
193	181	5.693878	78.44898	0.413265
186	175	-9.94898	3.44898	28.69898
181	185	-14.5918	9.877551	21.55612
188	175	-20.6633	14.87755	28.69898
184	177	0.479592	0.020408	11.27041
184	190	-1.37755	0.020408	92.98469
178	183	-16.2347	37.73469	6.984694
184	174	0.908163	0.020408	40.41327
		28.28571	369.7143	347.2143

Mx =184.1429 and My = 180.3571

28.28517 r = 369.143 * 347.2143

r = 0.078947

linear correlation coefficient  r = 0.079

Given claim is Determine whether there is sufficient evidence to support a claim of correlation between two variable .

Test of Hypothesis :

H₀ : $\rho$ = 0 versus H₁ : $\rho$ $\neq$ 0

Test Statistic :

$t =\frac{r*\sqrt{n-2}}{\sqrt{1-r^2}}$

t= (0.079*$\sqrt{14-2}$ )/$\sqrt{1-0.079^2 }$

t = 0.27

p - value : Under H0 t follows t distribution with n-2 df

$\alpha$ = 0.05

p - value = 2*P(t > 0.27) = 0.791747.

p - value = 0.792

Since p - value > $\alpha$ = 0.05 then we Fail to reject H0 at 0.05 significance level

Because the p value of the liner correlation coefficient is greater than the significance level , there is not sufficient evidence to support the claim that there is linear correlation between the height of winning presidential candidates and the height of there Opponent .