On the basis of these data and since our calculated P-value is less than 0.05, there is sufficient evidence to support the claim that the new program significantly reduces the recovery time.
Average time spent by patients in the hospital recovering fromm an appendix operation = 6.3 days
The sample mean = 1.2 days
The average time spent by 10 patients using the new recovery program = 5.5. days
Level of significance = 0.05
This is a case of a one-sample t-test. We compare the sample mean (5.5 days) to the population mean (6.3 days).
The null hypothesis (H₀) and alternative hypothesis (Hₐ) are as follows:
Null Hypothesis (H₀): The mean recovery time under the new program is equal to 6.3 days.
Alternative Hypothesis (Hₐ): The mean recovery time under the new program is less than 6.3 days.
The test statistic for a one-sample t-test is calculated as follows:
where:
is the sample mean,
μ is the population mean,
s is the standard deviation of the sample, and
n is the sample size.
Substituting the values into the formula:
= 2.12
Thus, the test statistic is 2.12.
The degrees of freedom in this case is 10 - 1 = 9.
The P-value associated with this t-statistic can be found using a t-distribution table or a statistical software and given a t-statistic of -2.12 with 9 degrees of freedom and a one-tailed test, the P-value would be less than 0.05.
If the P-value is less than the significance level (0.05), we reject the null hypothesis.
Thus, since our calculated P-value is less than 0.05, there is sufficient evidence to support the claim that the new program significantly reduces the recovery time.