Question

A study is done to determine if students in the California state university system take longer to graduate than students enrolled in private universities. 100 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 1.5591 years and 1 year, respectively. Suppose 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 0.3. Conduct a hypothesis test at the 5% level.

Answer 1

Answer:

There is enough evidence to support the claim that students in the California state university system take significantly longer to graduate than students enrolled in private universities (P-value = 0.0007).

Step-by-step explanation:

This is a hypothesis test for the difference between populations means.

The claim is that students in the California state university system take significantly longer to graduate than students enrolled in private universities.

Then, the null and alternative hypothesis are:

$H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2> 0$

The significance level is α=0.05.

The sample standard deviation is take into account and performed a two sample t-test. The 100 students are considered to be divided equally between state and private.

The sample 1 (state), of size n1=50 has a mean of 4.5 and a standard deviation of 0.8.

The sample 2 (private), of size n2=50 has a mean of 4.1 and a standard deviation of 0.3.

The difference between sample means is Md=0.4.

$M_d=M_1-M_2=4.5-4.1=0.4$

The estimated standard error of the difference between means is computed using the formula:

$s_{M_d}=\sqrt{\dfrac{\sigma_1^2+\sigma_2^2}{n}}=\sqrt{\dfrac{0.8^2+0.3^2}{50}}\\\\\\s_{M_d}=\sqrt{\dfrac{0.73}{50}}=\sqrt{0.015}=0.121$

Then, we can calculate the t-statistic as:

$t=\dfrac{M_d-(\mu_1-\mu_2)}{s_{M_d}}=\dfrac{0.4-0}{0.121}=\dfrac{0.4}{0.121}=3.3104$

The degrees of freedom for this test are:

$df=n_1+n_2-1=50+50-2=98$

This test is a right-tailed test, with 98 degrees of freedom and t=3.3104, so the P-value for this test is calculated as (using a t-table):

$\text{P-value}=P(t>3.3104)=0.0007$

As the P-value (0.0007) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that students in the California state university system take significantly longer to graduate than students enrolled in private universities.