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 1.5231 years and 1 year, respectively. The following data are collected. The California state university system students took on average 4.6 years with a standard deviation of 0.8. The private university students took on average 4.2 years with a standard deviation of 0.3. Conduct a hypothesis test at the 5% level. NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

-state the null hypothesis

-state the alternative hypothesis

-In words, state what your random variable X_state − X_private  represents.

-State the distribution to use for the test. (Round your answers to two decimal places.)

X_state − X_{private ~ __ ( __ , __ )}

_-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.)
-What is the p-value? (Round your answer to four decimal places.)
-Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. (Upload your file below.)

-(i) Alpha (Enter an exact number as an integer, fraction, or decimal.)
α =

-decision rejected or or do not reject

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

a) Null hypothesis: X_state = X_private
b) Alternative hypothesis: X_state > X_private

c) X_state − X_private represents the mean difference between the time taken to graduate for California state and University and private University.

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample z-test of the null hypothesis.

d) X_state − X_private~ Normal ( 0.40, 0.85)

Analyze sample data. Using sample data, we compute the standard error (SE), z statistic test statistic (z).

SE = 0.08544
z = [ (x₁ - x₂) - d ] / SE

z = 4.68

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is thesize of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.

The observed difference in sample means produced a t statistic of 4.68.

P-value = P(t > 4.68)

Use the t-calculator to determine the p-value

P-value = 0.00

α = 0.05

Interpret results. Since the P-value (0.00) is less than the significance level (0.05), we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that California state university system take longer to graduate, on average, than students enrolled in private universities.