Long term experience showed that after a type of eye surgery, recovery time has a mean of u = 5.3 days in a hospital. A random sample of 40 patients, with this type of eye surgery, were treated as outpatients during recovery and the sample mean recovery time was 4.2 days. Does this indicate that the mean recovery time for outpatients is less than the mean time for those recovering in the hospital? Use a 1% level of significance and assume o = 1.9 days.
To determine if the mean recovery time for outpatients is less than the mean time for those recovering in the hospital, we can perform a hypothesis test.
The null hypothesis (H0) states that there is no significant difference between the mean recovery time for outpatients and the mean recovery time in the hospital. The alternative hypothesis (Ha) states that the mean recovery time for outpatients is less than the mean recovery time in the hospital.
H0: μ (outpatients) = μ (hospital)
Ha: μ (outpatients) < μ (hospital)
We can use a one-sample t-test to compare the sample mean recovery time for outpatients to the known mean recovery time in the hospital. The formula for the t-test is:
t = (x̄ - μ) / (σ / √n)
where:
x̄ = sample mean recovery time for outpatients
μ = population mean recovery time in the hospital
σ = population standard deviation (assumed to be 1.9 days)
n = sample size
Given the information provided:
x̄ = 4.2 days
μ = 5.3 days
σ = 1.9 days
n = 40 patients
Let's calculate the t-value:
t = (4.2 - 5.3) / (1.9 / √40)
t = -1.1 / (1.9 / √40)
t ≈ -2.39
To calculate the critical t-value at a 1% level of significance, we need to look up the value in the t-distribution table. Since we want the left-tail value, we need to find the t-value with 1% area to the left and degrees of freedom (df) equal to n - 1.
df = 40 - 1 = 39
Looking up the t-value in the table, we find that the critical t-value at a 1% level of significance with 39 degrees of freedom is approximately -2.70.
Since the calculated t-value (-2.39) does not exceed the critical t-value (-2.70), we do not have sufficient evidence to reject the null hypothesis. Therefore, we cannot conclude that the mean recovery time for outpatients is less than the mean time for those recovering in the hospital at a 1% level of significance.