Question

A study is done to determine if students in the California state university system take longer to graduate, on average, than students enrolled in private universities. One hundred students from both the California state university system and private universities are surveyed. Suppose that from years of research, it is known that the population standard deviations are 1.5131 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 0.5. The private university students took on average 4.1 years with a standard deviation of 0.2. Conduct a hypothesis test at the 5% level. Xstate − Xprivate represents the difference in the average amount of time needed to graduate from a state university and a private university.

A. State the distribution to use for the test. (Round your answers to two decimal places.) Xstate − Xprivate ~ N = ( ? , ? )

B. What is the test statistic? (If using the z distribution round your answer to two decimal places, and if using the t distribution round your answer to three decimal places.)

C. What is the p-value? (Round your answer to four decimal places.)

D. Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. (insert below, thanks!!)

Answer 1

Solution :

Since, sample size is 100, we will use z-test .

And the underlying population is assumed to be normal.

X̅_state - X̅_private ~ N(X̅_state - X̅_private, SD²_state + SD²_private-2*Cov(X̅_state - X̅_private)

X̅_state - X̅_private ~ N( 4.5-4.1, 0.5² + 0.2² -2*0) #Cov is zero because they both are independent.

X̅_state - X̅_private ~ N(0.4, 0.29)

b)

Test statistic :

z = (X̅_state - X̅_private)/ √(σ²_state / n + σ²_private /n )

z= 0.4/√(1.5131² + 1²/100) #Using the known standard deviation from the question.

z= 2.205

Z = 2.21

p-value: 0.0135526

P Value = 0.0136