Question

A student at a four-year college claims that average enrollment at four-year colleges is higher than at two-year colleges in the United States. Two surveys are conducted. Of the 35 two-year colleges surveyed, the average enrollment was 5063 with a standard deviation of 4774. Of the 35 four-year colleges surveyed, the average enrollment was 5216 with a standard deviation of 8131. Conduct a hypothesis test at the 5% level.

Answer 1

Answer:

$t=\frac{(5216-5063)-0}{\sqrt{\frac{8131^2}{35}+\frac{4774^2}{35}}}}=0.096$  

The p value for this case would be given by:

$p_v =P(t_{68}>0.096)=0.92$  

Since the p value is higher than the significance level we don't have enough evidence to conclude that mean for 4 year college is higher than the mean for the 2 year college

Step-by-step explanation:

Data given

$\bar X_{1}=5063$ represent the mean for the two year college

$\bar X_{2}=5216$ represent the mean for the 4 year college

$s_{1}=4774$ represent the sample standard deviation for the two year college

$s_{2}=8131$ represent the sample standard deviation for the 4 year college

$n_{1}=35$ sample size for the group 1

$n_{2}=35$ sample size for the group 2  

$\alpha=0.05$ Significance level provided

t would represent the statistic

System of hypothesis

We want to verify if average enrollment at four-year colleges is higher than at two-year colleges in the United States, the system of hypothesis would be:  

Null hypothesis:$\mu_{1}-\mu_{2}\leq 0$  

Alternative hypothesis:$\mu_{2} - \mu_{1}> 0$  

Since we don't know the population deviations the statisitc is given by:

$t=\frac{(\bar X_{2}-\bar X_{1})-\Delta}{\sqrt{\frac{s^2_{1}}{n_{1}}+\frac{s^2_{2}}{n_{2}}}}$ (1)  

And the degrees of freedom are given by $df=n_1 +n_2 -2=35+35-2=68$  

Replacing the info given we got:

$t=\frac{(5216-5063)-0}{\sqrt{\frac{8131^2}{35}+\frac{4774^2}{35}}}}=0.096$  

P value  

The p value for this case would be given by:

$p_v =P(t_{68}>0.096)=0.92$  

Since the p value is higher than the significance level we don't have enough evidence to conclude that mean for 4 year college is higher than the mean for the 2 year college