Question

How do you construct a confidence interval for a correlation coefficient?

Answer 1

Hint: For solving this question you should know about the construct confidence interval of correlation coefficient. In this question first we will find the correlation coefficient and then we make the confidence interval. And there are three types of correlation coefficient which are; Positive Correlation, Negative correlation and No correlation.

Complete step-by-step solution:
According to our question it is asked to construct a confidence interval for a correlation coefficient.
Generally, Correlation coefficients are used to measure how strong a relationship is between two variables. And many types of correlation coefficient are available there but the main is Pearson’s correlation. Pearson's correlation is commonly used in linear regression.
The formula for correlation coefficient is:
\[r=\dfrac{\sum{\left( {{x}_{i}}-\overline{x} \right)\left( {{y}_{i}}-\overline{y} \right)}}{\sqrt{\sum{{{\left( {{x}_{i}}-\overline{x} \right)}^{2}}\sum{{{\left( {{y}_{i}}-\overline{y} \right)}^{2}}}}}}\]
Where, r = correlation coefficient
\[{{x}_{i}}\] = Values of the x – variable in a sample
\[\overline{x}\] = mean of the values if the x – variable
\[{{y}_{i}}\] = Values of y – variable in a sample
\[\overline{y}\] = mean of the values of y – variable
Assuming a bivariate normal population with population correlation \[\rho \], the transformation of the sample product moment correlation from r to \[{{z}_{r}}\]:
\[{{z}_{r}}=\dfrac{1}{2}\ln \left( \dfrac{1+r}{1-r} \right)\]
is approximately normally distributed with variance \[\dfrac{1}{\left( n-3 \right)}\]. The lower and upper confidence limits for \[\rho \] are obtained by
\[{{z}_{r}}\pm {{z}_{1-{\alpha }/{2}\;}}\sqrt{\dfrac{1}{\left( n-3 \right)}}\]
To obtain \[{{z}_{L}}\] and \[{{z}_{u}}\], the values of \[{{z}_{L}}\] and \[{{z}_{u}}\] are than transformed back to the correlation scale using the inverse transformations.
\[{{r}_{L}}=\dfrac{\exp \left( 2{{z}_{L}} \right)-1}{\exp \left( 2{{z}_{L}} \right)+1}\]
And
\[{{r}_{u}}=\dfrac{\exp \left( 2{{z}_{u}} \right)-1}{\exp \left( 2{{z}_{L}} \right)+1}\]
One sided limit may be obtained by replacing \[{}^{\alpha }/{}_{2}\] by \[\alpha \].
Or if we want to calculate the sample size of a two-sided interval when W has been specified is \[W={{r}_{u}}-{{r}_{L}}\].
For one sided intervals, the distance from the sample correlation to limit, D is specified.
The basic equation for determining sample size for a one – sided upper limit when D has been specified as –
\[D={{r}_{u}}-r\]
And one-sided lower limit when D has been specified as:
\[D=r-{{r}_{L}}\]

Note: If you want to construct a confidence interval for a correlation coefficient then you should be careful during counting the correlation coefficient. And find all the upper and lower limits of the confidence interval and calculate these at the specified W and D. And if one side limit is calculating then also use the specified D.