Question

A study is done to determine if students in the California state university system (sample 1) take longer to graduate, on average, than students enrolled in private universities (sample 2). Suppose that from years of research, it is known that the population standard deviations are 1.5 years and 0.9 years, respectively. One hundred students from each the California state university system and private universities are surveyed. The California state university system students took on average 4.8 years while the private university students took on average 4.4 years. Conduct a hypothesis test, using =0.10.

Answer 1

Answer:

We conclude that the students in the California state university system take longer to graduate as compared to students enrolled in private universities.

Step-by-step explanation:

We are given that from years of research, it is known that the population standard deviations are 1.5 years and 0.9 years, respectively.

One hundred students from each the California state university system and private universities are surveyed. The California state university system students took on average 4.8 years while the private university students took on average 4.4 years.

Let $\mu_1$ = population average time taken by students in the California state university system to graduate

     $\mu_2$ = population average time taken by students in the private universities to graduate

SO, Null Hypothesis, $H_0$ : $\mu_1-\mu_2\leq0$  or  $\mu_1 \leq \mu_2$   {means that the students in the California state university system take less than or equal time to graduate as compared to students enrolled in private universities}

Alternate Hypothesis, $H_A$ : $\mu_1-\mu_2 >0$  or  $\mu_1>\mu_2$   {means that the students in the California state university system take longer to graduate as compared to students enrolled in private universities}

The test statistics that will be used here is Two-sample z test statistics as we know about the population standard deviations;

                         T.S.  = $\frac{(\bar X_1-\bar X_2)-(\mu_1-\mu_2)}{\sqrt{\frac{\sigma_1^{2} }{n_1} + \frac{\sigma_2^{2} }{n_2}} }$  ~ N(0,1)

where,  $\bar X_1$ = sample average time taken by the California state university system students to graduate = 4.8 years

$\bar X_2$ = sample average time taken by the private university students to graduate = 4.4 years

$\sigma_1$ = population standard deviation for California state university system students = 1.5 years

$\sigma_2$ = population standard deviation for private university students = 0.9 years

$n_1$ = sample of California state university system students taken = 100

$n_2$ = sample of private university students taken = 100

So, test statistics  =   $\frac{(4.8-4.4)-(0)}{\sqrt{\frac{1.5^{2} }{100} + \frac{0.9^{2} }{100}} }$

                               =  2.287

Now at 0.10 significance level, the z table gives critical value of 1.2816 for right-tailed test. Since our test statistics is more than the critical value of z so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.

Therefore, we conclude that the students in the California state university system take longer to graduate as compared to students enrolled in private universities.