A pair of Null and Alternative Hypothesis is formulated as follows for the statistical test:
H0 : The ratio of man biking at night over man biking during the day is the same or less than the ratio of woman biking at night to woman biking during the day. \(\frac{Wnight}{Wday}\ge\frac{Mnight}{Mday}\)
H1: The ratio of man biking at night over man biking during the day is higher than the ratio of woman biking at night to woman biking during the day. \(\frac{Wnight}{Wday}<\frac{Mnight}{Mday}\)
Since the number of observations in the sample is much larger than 30 and the null hypothesis is formulated in the form of proportion, the Z-test for two proportions is used to test the hypothesis. The mathematical formula for computing the statistics are listed as follows:
\(Z\ score:\ Z\ =\ \frac{p0\ -\ p1}{SE}\)
\(Where:\ p0\ =\ \frac{Wnight}{Wday},\ p1\ =\ \frac{Mnight}{Mday},\ SE\ =\ \sqrt{p\left(1-p\right)\left(\frac{1}{n0}\ -\ \frac{1}{n1}\right)}\ ,\ p\ =\ \frac{p0n0\ +\ p1n1}{n0\ +\ n1}\)\(Z\ score:\ Z\ =\ \frac{p0\ -\ p1}{SE}\)