The Medical Rehabilitation Foundation reports that the average cost of rehabilitation for stroke victims is at least $24,500. To see if the average cost might be different at a particular hospital, a researcher selects a random number sample of 50 stroke victims and finds that the average cost is $25,225. The standard deviation of the population os $1,300. At a = .01, can it be concluded that the average cost of stroke rehabilitation at a particular hospital is at least $24,500?
To determine if it can be concluded that the average cost of stroke rehabilitation at a particular hospital is at least $24,500, we can perform a hypothesis test.
Let's set up the null and alternative hypotheses:
Null Hypothesis (H0): The average cost of stroke rehabilitation at a particular hospital is $24,500.
Alternative Hypothesis (Ha): The average cost of stroke rehabilitation at a particular hospital is greater than $24,500.
Next, we need to determine the test statistic and the critical value to compare it with. For this case, we will use a t-test because the population standard deviation is unknown, and the sample size is relatively small (n = 50).
The formula for the t-test statistic is:
t = (sample mean - population mean) / (sample standard deviation / √n)
Given information:
Sample mean (x̄) = $25,225
Population mean (μ) = $24,500
Sample standard deviation (s) = $1,300
Sample size (n) = 50
Significance level (α) = 0.01 (1% confidence level)
First, we need to calculate the t-statistic using the formula:
t = (25,225 - 24,500) / (1,300 / √50)
t = 725 / (1,300 / 7.071)
t ≈ 725 / 183.83
t ≈ 3.95
Next, we need to find the critical value (tc) for a one-tailed test with α = 0.01 and 49 degrees of freedom (n - 1). We can consult a t-distribution table or use statistical software to find this value. For this example, the critical value is approximately 2.685.
Finally, we compare the calculated t-statistic to the critical value. If the calculated t-value is greater than the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Since the calculated t-statistic (3.95) is greater than the critical value (2.685), we reject the null hypothesis.
Therefore, at a significance level of 0.01, we can conclude that the average cost of stroke rehabilitation at the particular hospital is significantly higher than $24,500.