Question

95. A group of transfer bound students wondered if they will spend the same mean amount on texts and supplies each year at their four-year university as they have at their community college. They conducted a random survey of $54$ students at their community college and $66$ students at their local four-year university. The sample means were $\backslash (947$ and $\backslash )1,011$, respectively. The population standard deviations are known to be $\backslash (254$and $\backslash )87$, respectively. Conduct a hypothesis test to determine if the means are statistically the same.

Short answer

Subscripts :$1=$community college, $2=$ four - year university
(a) The null hypothesis: $H_0:{\mu }_1={\mu }_2$
(b) The alternate hypothesis: $H_a:{\mu }_1\ne {\mu }_2$
(c) The random variable is the difference between the average text costs and each year's supplies at community institutions, divided by four.
(d) Normal distribution.
(e) The test statistic: $1.768$
(f)The $p$-value: $0.0769$
(g) Check student's solution.
(h) (i) Alpha: $0.05$
(ii) Decision: not reject the null hypothesis.
(iii) Reason for Decision: $p$-value $>\alpha$
(iv) As a result, there is not enough proof to conclude that the mean costs of texts and supplies at community colleges and $4$-year universities is different at the $5\%$significance level.

Step-by-step solution

Given information

To conduct a hypothesis test, to determine if the means are statistically the same.

Explanation

Subscripts : $1=$community college, $2=$four - year university
(a) The null hypothesis is described as follows:
$H_0:{\mu }_1={\mu }_2$
(b) The alternate hypothesis is described as follows:
$H_a:{\mu }_1\ne {\mu }_2$
(c)The random variable is the difference between the average text costs and each year's supplies at community institutions, divided by four.
(d) Normal distribution.
(e) To determine the test static as:
To access the stat list editor, click STAT and then 1.

Then ENTER all the values the OUTPUT will be:

Hence, the test statistics is $1.768$.

Explanation

(f) The $p$-value from the output is determined as $0.0769$.
(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.

(h)

(i) $\alpha =0.05$
(ii) Decision: not reject the null hypothesis.
(iii) Reason for Decision: $p$- value$>\alpha$.
(iv) As a result, there is not enough proof to conclude that the mean costs of texts and supplies at community colleges and $4$-year universities is different at the $5\%$level of significance.