Question

93. 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 1.5811 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 0.3.

Short answer

(a) The null hypothesis: $H_0:{\mu }_1\le {\mu }_2$
(b) The alternate hypothesis: $H_a:{\mu }_1>{\mu }_2$
(c) The random variable is the difference between the mean time it takes to graduate from a California public institution and a private university.
(d) Normal distribution.
(e) The test statistics: $2.138$.
(f) The $p$-value is $0.01625$.
(g) (i) $\alpha =0.05$
(ii) Decision: Reject null when $\alpha =0.05$; do not reject null when $\alpha =0.01$.
(iii) Reason for Decision: $p$-value $<\alpha$.
(iv) As a result, there is enough proof to conclude that the mean time it takes to graduate from California state universities is more than that of private universities at the $5\%$level of significance.

Step-by-step solution

Given information

Let, the California state university system students took on average $4.5$years with a standard deviation of $0.8$. And the private university students took on average $4.1$years with a standard deviation of $0.3$.

Explanation

(a) The null hypothesis is described as follows:
$H_0:{\mu }_1\le {\mu }_2$
(b) The alternate hypothesis is described as follows:
$H_a:{\mu }_1>{\mu }_2$
(c) The random variable is the difference between the mean time it takes to graduate from a California public institution and a private university.
(d) Normal distribution.
(e) To determine test static as:
To enter the stat list editor, press STAT followed by $1$.

Then ENTER all the values will be:

Hence the test statistics is $2.138$.

Explanation

(f) The $p$-value from the output is determined as $0.01625$.
(g) Obtain a clear picture of the situation using the information from the last task.
The horizontal axis should be named and measured, and the region(s) on the graph that correlate to the$p$- value should be shaded.

(i) $\alpha =0.05$
(ii) Decision: Reject null when $\alpha =0.05$; do not reject null when $\alpha =0.01$.
(iii) Reason for Decision: $p-$value $<\alpha$.
(iv) As a result, there is enough proof to conclude that the mean time it takes to graduate from California state universities is more than that of private universities, at the $5\%$level of significance, .