A study is done to determine if students in the California state university system take longer to graduate than students enrolled in private universities. One hundred students from both the California state university system and private universities are surveyed. From years of research, it is known that the population standard deviations are 1.5811 years and 1 year, respectively. The following data are collected.

The California state university system students took on average 4.5 years with a standard deviation of 0.8.
The private university students took on average 4.1 years with a standard deviation of 0.2.

Required:
Conduct a hypothesis test at the 5% level.

[tex]\overline X _{state} - \overline X_{private} \sim N(\overline X _{state}-\overline X _{private, SD^2_{state} + SD^2_private} -2\times Cov(\overline X _{state} - \overline X_{private}})}[/tex]

[tex]\overline X _{state} - \overline X_{private} \sim N(4.5-4.1, 0.8^2 + 0.2^2 -2\times 0})}[/tex]

[tex]\overline X _{state} - \overline X_{private} \sim N(0.4, 0.68})}[/tex]

The test statistics:

[tex]z = \dfrac{\overline X_{state} - \overline X_{private} }{ \sqrt{\dfrac{\sigma^2_{state}}{n } + \dfrac{\sigma^2_{private}}{n } } }[/tex]

[tex]z = \dfrac{0.4 }{ \sqrt{\dfrac{1.5811^2}{100 } + \dfrac{1^2}{100 } } }[/tex]

z = 2.138

Using the z tables;

P-value = (Z> 2.138)

P-value = 1 - (Z<2.138)

P-value = 1 - 0.9837

P-value = 0.0163

Decision rule: to reject the null hypothesis if the p-value is less than the significance level

Conclusion: We reject the null hypothesis and conclude that there is enough evidence to conclude that the average time it requires for the students to graduate from a private university is lesser than that of the time it takes such student to graduate from the California state university system.