A company claims that cancer patients using drug X will live at least 8 more years.
n=25
= 8.5 year
s = 1.5 year
a) Test at a=.01
b) Give the CIE at 95%
To answer these questions, we need to perform hypothesis testing and calculate the confidence interval.
a) Test at a = 0.01:
To test the claim made by the company, we set up the null and alternative hypotheses as follows:
Null hypothesis (H0): The mean survival time of cancer patients using drug X is equal to 8 years.
Alternative hypothesis (Ha): The mean survival time of cancer patients using drug X is greater than 8 years.
To perform the test, we use the z-test since we know the population standard deviation (s). The test statistic can be calculated using the formula:
z = (x̄ - μ) / (s / √n)
Where:
x̄ is the sample mean
μ is the population mean
s is the population standard deviation
n is the sample size
In this case, x̄ = 8.5 years, μ = 8 years, s = 1.5 years, and n = 25. Plugging these values into the formula, we get:
z = (8.5 - 8) / (1.5 / √25)
z = 0.5 / (1.5 / 5)
z = 0.5 / 0.3
z = 1.67
To determine the critical value for a significance level of 0.01, we look up the z-value in the standard normal distribution table. The critical z-value for a one-tailed test at α = 0.01 is approximately 2.33.
Since the calculated test statistic (1.67) is less than the critical value (2.33), we fail to reject the null hypothesis. Therefore, there is not enough evidence to conclude that the mean survival time of cancer patients using drug X is greater than 8 years.
b) Calculate the 95% confidence interval (CI):
To calculate the confidence interval, we use the formula:
CI = x̄ ± z * (s / √n)
Where:
x̄ is the sample mean
z is the z-value corresponding to the desired confidence level
s is the population standard deviation
n is the sample size
For a 95% confidence interval, the z-value is approximately 1.96 (obtained from the standard normal distribution table). Plugging in the values, we get:
CI = 8.5 ± 1.96 * (1.5 / √25)
CI = 8.5 ± 1.96 * (1.5 / 5)
CI = 8.5 ± 1.96 * 0.3
CI = 8.5 ± 0.588
The 95% confidence interval for the mean survival time of cancer patients using drug X is approximately (7.91, 9.09). This means that we are 95% confident that the true mean survival time falls within this range.