Answer:
We conclude that the mean cost-to-charge ratio for Oregon hospitals is lower for outpatient care than for inpatient care.
Step-by-step explanation:W
We are given with the cost-to-charge ratios for both inpatient and outpatient care in 2002 for a sample of six hospitals in Oregon below;
Hospital       2002 Inpatient Ratio         2002 Outpatient Ratio
     1                           68                                            54
     2                          100                                           75
     3                           71                                             53
     4                           74                                            56
     5                          100                                           74
     6                           83                                            71
Let  = <u><em>mean cost-to-charge ratio for outpatient care</em></u>
 = <u><em>mean cost-to-charge ratio for outpatient care</em></u>
 = <u><em>mean cost-to-charge ratio for impatient care</em></u>.
 = <u><em>mean cost-to-charge ratio for impatient care</em></u>.
SO, Null Hypothesis,  :
 :  {means that the mean cost-to-charge ratio for Oregon hospitals is higher or equal for outpatient care than for inpatient care}
     {means that the mean cost-to-charge ratio for Oregon hospitals is higher or equal for outpatient care than for inpatient care}
Alternate Hypothesis,  :
 :  {means that the mean cost-to-charge ratio for Oregon hospitals is lower for outpatient care than for inpatient care}
     {means that the mean cost-to-charge ratio for Oregon hospitals is lower for outpatient care than for inpatient care}
The test statistics that would be used here is <u>Two-sample t-test statistics</u> because we don't know about population standard deviations;
                          T.S. =   ~
  ~ 
where,  = sample mean cost-to-charge Outpatient Ratio =
 = sample mean cost-to-charge Outpatient Ratio =  = 63.83
 = 63.83
 = sample mean cost-to-charge Impatient Ratio =
 = sample mean cost-to-charge Impatient Ratio =  = 82.67
 = 82.67
 = sample standard deviation for Outpatient Ratio =
 = sample standard deviation for Outpatient Ratio =  = 10.53
 = 10.53
 = sample standard deviation for Impatient Ratio =
 = sample standard deviation for Impatient Ratio =  = 14.33
 = 14.33
 = sample of hospital for outpatient care = 6
 = sample of hospital for outpatient care = 6
 = sample of hospital for outpatient care = 6
 = sample of hospital for outpatient care = 6
Also,  =
 =   = 12.574
 = 12.574
So, <u><em>the test statistics</em></u>  =   ~
  ~ 
                                      =  -2.595
The value of t test statistics is -2.595.
<u>Now, at 5% significance level, the t table gives critical value of -1.812 at 10 degree of freedom for left-tailed test.</u>
Since, our test statistics is less than the critical value of t as -2.595 < -1.812, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which <u><em>we reject our null hypothesis</em></u>.
Therefore, we conclude that the mean cost-to-charge ratio for Oregon hospitals is lower for outpatient care than for inpatient care.