. Each patient at a hospital is asked to evaluate the service at the time of discharge. Recently there have been several complaints that resident physicians and nurses on the surgical wing respond too slowly to the emergency calls of senior citizens compared to other patient groups. The administrator of the hospital asked the quality assurance department to investigate. After studying the problem, the quality assurance department collected the following sample information.
Patient type
Smaple mean
Sample standard deviation
Sample Size
Senor Citizens
5.5 Minutes
0.40 minuets
50
Other
5.3 Minutes
0.30 minutes
100
Required: At the 0.01 significance level, is the response time longer for the senior citizens, emergencies?
A. Conduct hypothesis testing for the differences in the population means based on the two independent samples.
B. What is the p-value in this problem? Interpret it.
A. To conduct hypothesis testing for the differences in the population means based on the two independent samples, we can perform a two-sample t-test.
The null hypothesis (H0) states that there is no difference in the response time between senior citizens and other patient groups. The alternative hypothesis (Ha) states that the response time for senior citizens is longer than other patient groups.
Assuming equal variances, the test statistic can be calculated using the formula:
t = (x1 - x2) / sqrt((s1^2/n1) + (s2^2/n2))
Where:
x1 = sample mean of senior citizens
x2 = sample mean of other patients
s1 = sample standard deviation of senior citizens
s2 = sample standard deviation of other patients
n1 = sample size of senior citizens
n2 = sample size of other patients
Plugging in the values from the sample information:
t = (5.5 - 5.3) / sqrt((0.4^2/50) + (0.3^2/100))
You can now calculate the t-value.
Next, we need to find the critical t-value for a two-tailed test at a significance level of 0.01 and degrees of freedom (df) equal to the smaller of (n1-1) and (n2-1).
Finally, compare the calculated t-value with the critical t-value. If the calculated t-value is greater than the critical t-value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
B. The p-value in this problem can be found by using the t-distribution table or calculator. It represents the probability of obtaining the observed difference (or more extreme) between the response times of the two groups, assuming the null hypothesis is true.
Interpreting the p-value, if it is less than the significance level (0.01 in this case), we reject the null hypothesis. This means that there is strong evidence to suggest that the response time for senior citizens is longer compared to other patient groups. If the p-value is greater than the significance level, we fail to reject the null hypothesis, indicating that there is not enough evidence to conclude a significant difference in response times.
A. To conduct hypothesis testing for the differences in the population means based on the two independent samples, we can use the two-sample t-test.
The null hypothesis (H0) states that the response time is the same for senior citizens and other patients in emergencies. The alternative hypothesis (Ha) states that the response time is longer for senior citizens in emergencies.
We can set up the following hypotheses:
H0: μ1 - μ2 = 0 (There is no difference in response time between senior citizens and other patients in emergencies)
Ha: μ1 - μ2 > 0 (Response time is longer for senior citizens in emergencies)
We can calculate the test statistic using the formula:
t = (x1 - x2) / sqrt((s1^2/n1) + (s2^2/n2))
Where:
x1 = sample mean for senior citizens
x2 = sample mean for other patients
s1 = sample standard deviation for senior citizens
s2 = sample standard deviation for other patients
n1 = sample size for senior citizens
n2 = sample size for other patients
Calculating the test statistic:
t = (5.5 - 5.3) / sqrt((0.4^2/50) + (0.3^2/100))
t = 0.2 / sqrt((0.016/50) + (0.009/100))
t = 0.2 / sqrt(0.00032 + 0.00009)
t = 0.2 / sqrt(0.00041)
t ≈ 2.68
Next, we need to find the critical value for a one-tailed test at the 0.01 significance level. Using a t-distribution table or a statistical calculator, we find the critical value to be approximately 2.61 (for degrees of freedom = 50 + 100 - 2 = 148).
Since the test statistic (2.68) is greater than the critical value (2.61), we reject the null hypothesis. This means that at the 0.01 significance level, there is evidence to suggest that the response time is longer for senior citizens in emergencies.
B. The p-value in this problem is the probability of observing a test statistic as extreme as the one calculated (t = 2.68), under the assumption that the null hypothesis is true. In other words, it is the probability of obtaining these sample results if there is no actual difference in the response time between senior citizens and other patients in emergencies.
To interpret the p-value, we compare it to the significance level (α = 0.01). If the p-value is less than or equal to the significance level, we reject the null hypothesis. In this case, the p-value is not provided, but we know that since the test statistic (2.68) is greater than the critical value (2.61), the p-value would be less than 0.01.
Therefore, we can interpret it as follows: The p-value is less than the significance level of 0.01, indicating strong evidence to suggest that the response time is longer for senior citizens in emergencies.
To test whether the response time is longer for senior citizens compared to other patient groups, we can conduct a hypothesis test based on the two independent samples.
Step 1: Set up the hypotheses:
- Null Hypothesis (H0): The response time for senior citizens is not longer than for other patient groups. (μ1 ≤ μ2)
- Alternative Hypothesis (H1): The response time for senior citizens is longer than for other patient groups. (μ1 > μ2)
Step 2: Determine the significance level:
The significance level (α) is given as 0.01, which means we are willing to accept a 1% chance of rejecting the null hypothesis when it is true.
Step 3: Compute the test statistic:
To compute the test statistic, we can use the formula for the two-sample t-test:
t = (x̄1 - x̄2) / sqrt((s1²/n1) + (s2²/n2))
Where:
x̄1 = sample mean for senior citizens = 5.5 minutes
x̄2 = sample mean for other patient groups = 5.3 minutes
s1 = sample standard deviation for senior citizens = 0.40 minutes
s2 = sample standard deviation for other patient groups = 0.30 minutes
n1 = sample size for senior citizens = 50
n2 = sample size for other patient groups = 100
Plugging in the values:
t = (5.5 - 5.3) / sqrt((0.40²/50) + (0.30²/100))
Step 4: Calculate the p-value:
To calculate the p-value, we need to look up the t-statistic in the t-distribution table or use a statistical software. The p-value represents the probability of observing a test statistic as extreme as the one calculated (or even more extreme) under the null hypothesis.
Step 5: Compare the p-value and significance level:
If the p-value is less than the significance level (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Interpreting the p-value:
If the p-value is smaller than 0.01 (the significance level), it means there is strong evidence to suggest that the response time for senior citizens is longer than for other patient groups. On the other hand, if the p-value is greater than 0.01, we do not have enough evidence to conclude that the response time for senior citizens is longer.